United States Department of Commerce

Technology Administration

National Institute of Standards and Technology

NIST Special Publication 866

Extreme Value Theory and Applications

Proceedings of the Conference on Extreme

Value Theory and Applications, Volume 3

Gaithersburg, Maryland, May 1993

Janos Galambos, James Lechner and Emil Simiu, Editors

Charles Hagwood, Technical Editor

NATL INST. OF STAND & TECH

^^^'^ REFERENCI

PUBLICATIONS

Figure 1 JacK-up plalfonn wiih cajililcvcr.

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Figure 10 Geomagnetic Field Contour Map On MOS-1 Orbit

'Vy\K \ (Cnimbcl). ,y( i ) = c\p(-c\|i(- v j). .v £ IR Type II U'n-chc/): = exp(-v~"), v > 0. a X)

Type III [Wcihiill): = e\p(-(-v)«), ,v < 0, a > 0

(JLSl

ft95

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1 1 n
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'At Boulder, CO 80303. 'Some elements at Boulder, CO 80303.
NIST Special Publication 860

Extreme Value Theory and Applications

Proceedings of the Conference on Extreme

Value Theory and Applications, Volume 3

Gaithersburg, Maryland, May 1993

Janos Galambos, James Lechner and Emil Simiu, Editors

Charles Hagwood, Technical Editor

Statistical Engineering Division

Computing and Applied Mathematics Laboratory

National Institute of Standards and Technology

Gaithersburg, MD 20899-0001

August 1994

U.S. Department of Commerce

Ronald H. Brown, Secretary

Technology Administration

Mary L. Good, Under Secretary for Technology

National Institute of Standards and Technology

Arati Prabhakar, Director
National Institute of Standards

and Technology

Special Publication 866

Natl. Inst. Stand. Technol.

Spec. Publ. 866

231 pages (Aug. 1994)

CODEN: NSPUE2

U.S. Government Printing Office

Washington: 1994

For sale by the Superintendent

of Documents

U.S. Government Printing Office

Washington, DC 20402
PREFACE

It appears that we live in an age of disasters: the Mississippi and the Missouri rivers flood

millions of acres, earthquakes bit Tokyo and California, airplanes crash due to mechanical

failure, and powerful windstorms cause increasingly costly damage. While these may seem to be unexpected phenomena to the man on the street, they are actually happening according to well defined rules of science known as extreme value theory. For many phenomena records

must be broken in the future, so if a design is based on the worst case of the past then we are not really prepared for the future. Materials will fail due to fatigue: even if the body of an

aircraft looks fine to the naked eye, it might suddenly fail if the aircraft has been in operation over an extended period of time. Extreme value theory has by now penetrated the social

sciences, the medical profession, economics, and even astronomy. We believe this field has

come of age. To utilize and stimulate progress in the theory of extremes and promote its application, an international conference was organized in which equal weight was given to theory and practice.

The Proceedings are published in three Volumes. Volume I, pubhshed by Klewer Academic

Publishers, contains papers of general interest in extreme value theory and practice. Volume II, a special issue of the NIST Journal of Research, contains papers deemed by the Committee to be most directly relevant to NIST's mission. Volume III (this volume) contains papers selected

for their important contribution to a number of specialized topics. All papers have been

refereed and we are grateful to the many scientists from all over the world for serving as

referees.

The conference was held on the campus of the National Institute of Standards and Technology

(NIST) in Gaithersburg, Maryland, with its Statistical Engineering Division (SED) acting as

host. It was organized by Temple University, Philadelphia, Pennsylvania, and NIST.

The conference had no external funding, and NIST's support was fundamental to its success. We are particularly grateful to Dr. Robert Lundegard, Chief of SED, whose support was the

single most important factor in making the conference happen. The support of NIST's Building

and Fire Research Laboratory is also acknowledged with thanks.

The Organizing Committee consisted of Janos Galambos (Chairman), James Lechner, Stefan

Leigh (Director of Local Arrangements), James Pickands III, Emil Simiu, and Grace Yang.

Stefan's enthusiasm and tireless work was essential for the success of the Conference.

The Conference included three special sessions:

The Centennial Session for Emil Gumbel. Churchill Eisenhart introduced the Session. His

personal recollections of Gumbel are included in Volume I of the Proceedings. Emil Simiu then

spoke on Gumbel's life and work.

iii
The Memorial Session for Josef Tiago de Oliveira. Janos Galambos remembered Tiago, a

close friend to many Conference participants, who was on the initial hst of invited speakers.

M. Ivette Gomes gave a detailed account of his work.

The 80th Birthday Session for B. V. Gnedenko. Janos Galambos

summarized the work of Gnedenko as the founder of modern extreme value theory and his

contributions to the central limit problem, hmit theorems with random sample size and renewal

theory.

Preceding the Conference, a Short Course was presented. Prof. Galambos gave an introductory

lecture on general principles of extreme value theory, and Prof. Castillo presented a four-hour course on "Engineering Analysis of Extreme Value Data." Prof. Castillo's notes were distributed

to all Conference participants.

The Conference was opened by Dr. Robert Lundegard who emphasized

extreme value theory's role in several scientific and engineering fields. It ended with a panel

discussion on the future of extreme value theory and its applications. The Panel was chaired by

Janos Galambos, and its members were Enrique Castillo, Laurens de Haan, Lucien Le Cam and

Richard L. Smith.

Finally, special thanks are extended to Shirley G. Bremer and Kaye Wade of the Statistical

Engineering Division at NIST, for their tireless and efficient work on preparation for the

Conference, including the typing of the Abstracts volume distributed at registration.

The Editors

iv
TABLE OF CONTENTS

On The Record Values From Univariate Distributions 1 Ahsanullah, M., Rider College, Lawerenceville, NJ

Composite Sampling And Extreme Values 7

Argon, E.D., Gore, S.D.,and Patil, G.P., The Pennsylvania State University,

University Park, PA

Extremal Sojourn Times For Markov Chains 19

Arnold, B.C., University of California, Riverside, CA

Bootstrapping Extremes Of I.I, D. Random Variables 23

Athreya, K.B., and Fukuchi, J., Iowa State University, Ames, lA

Extreme Analysis Of Wave Pressure And Corrosion

For Structural Life Prediction 31

Ayyub, B.M., University of Maryland, College Park, MD

Record Values From Rayleigh And Weibull Distributions

And Associated Inference 41

Balakrishnan, N., McMaster University, Hamilton, Ontario, Canada and

Chan, P.S., The Chinese University of Hong Kong, Shatin, Hong Kong

On The Estimation Of The Pareto Tail-Index

Using A;-Record Values 53

Berred, A.M., Universite du Havre, Le Havre Cedex, France

The Point-Process Approach To The Directional Analysis

Of Extreme Wind Speeds 63

Bortot, P., Universita di Padova, Padova, Italia

High Boundary Excursions Of Locally Stationary

Gaussian Processes 69

Braker, H. U., Institut fur Mathematische und Versicherungslehre,

Bern, Germany

Asymptotic Approximations For The Crossing Rates

Of Poisson Square Waves 75

Breitung, K., Sem. F.A. Stochastik, Akademiestr. 1/IV, Munich, Germany

V
Meso-Scale Estimation Of Expected Extreme Values 81

Burton, R.M., Goulet, M.R., and Yim, S.C.S., Oregon State

University, Corvallis, OR

An Expert System Prototype For The Analysis Of

Extreme Value Problems 85

Castillo, E., Alvarez, E., Coho, A. and Herrero, M.T., University of

Cantahria, Santander, Spain

Poisson Approximation Of Point Processes Of Exceedances

under von Mises Conditions 95

Drees, H., Universitdt zu Koln, Koln, Germany and Kaufmann, E.,

Universitdt-Gh Siegen, Siegen, Germany

Estimating The Extremal Index Under A Local

Dependence Condition By The Reciprocal Of The Average

Length Of Successive Runs 103

Duarte, L.C.C., University of Lisbon, Lisbon, Portugal

Approximate Extreme Value Analysis For A Rigid Block

Under Seismic Excitation Ill

Facchini, L., and Spinelli, P., University of Florence, Florence, Italy

The Rate Of Convergence Or Divergence For Percentiles Of Gamma

Distributions And Its Application To Sample Extremes 119

Gan, G., and Bain, L.J., University of Missouri, Rolla, MO

Application Of Extreme-Value Theory To Reliability Physics

Of Electronic Parts (On-Orbit Single Event Phenomena) 123

Goka, T., National Space Development Agency of Japan, Tokyo, Japan

Certain Identities In Expectations Of Functions Of

Order Statistics And Characterization Of Distributions 131

Govindarajulu, Z., University of Kentucky, Lexington, KY

Investigating The Bias And MSE Of Exceedance Based

Tail Estimators For The Cauchy Distribution 139

Grimshaw, S.D., Brigham Young University, Provo, UT

vi
Estimating Quantiles For A Type III Domain Of Attraction

Based On The k Largest Observations 149

Hasofer, A.M., The University of New South Wales, New South Wales,

Australia and Wang, J.Z., University of Western Sydney, New South Wales,

Australia

Extreme Values Of Monotonia Functions And Evaluation

Of Catastrophic Flood Loss 157

Lambert, J.H., Li, D. and Haimes, Y.Y., University of Virginia,

Charlotttesville, VA

Second Order Behavior Of Domains Of Attraction And The Bias

Of Generalized Pickands' Estimator 165

Pereira, T.T., University of Lisbon, Lisbon, Portugal

Normal Sample Range: Asymptotic Distribution, Approximations

And Power Comparisons 179

Rukhin, A.L., UMBC, Baltimore, MD

Estimation Of Extreme Sea Levels At Major Ports In Korea 187

Shim, J., Oh, B.C., and Jun, K.C., Korea Ocean Research &

Development Institute, Seoul, Korea

Limit Properties Of Maxima Of Weighted I.I.D. Random Variables . . . 197

Tomkins, R.J., University of Regina, Regina, Saskatchewan, Canada

Large Deviations For Order Statistics 203

Vinogradov, V., Concordia University, Montreal, Quebec, Canada

Extremes For Independent Nonstationary Sequences 211

Weissman, L, Technion-Israel Institute of Technology, Haifa, Israel

Order Statistics And Proofs Of Combinatorial Identities 219

Wenocur, R.S., University of Pennsylvania, Philadelphia, PA

An Examination Of The Extremes Of Selected New Zealand

Rainfall And Runoff Records For Evidence Of Trend 223

Withers, C.S., and Silby, W. W., Institute for Industrial Research Development,

Lower Hutt, New Zealand

Extreme Values In Business Interruption Insurance 231

Zajdenweber, D., Universite de Paris X, Nanterre Cedex, France

vii

On The Record Values From Univariate

Distributions

Ahsanullah, M.

Rider College, Lawerenceville, NJ

In this paper the basic concepts and properties ofthe records of univariate continuous distributions are presented. Inferences about the location and scale parameters of a class of univariate

distributions are given. Prediction of sth record value based on the observed first m ( m < s) record

values are discussed.

1.0 Introduction

Suppose that Xj, X2,... is a sequence of

independent and identically distributed (i.i.d.) random

variables with cumulative distribution function F(x). Set Yjj = max (min){X]^, ...,Xn\, n > 1. We say X; is an upper (lower)record value 01 { Xj^, n> 1}, if Vj >

(<)Yj.]^, j>l. By definition, Xj is an upper as well

as a lower record value. Thus the upper record values

in the sequence (X^ ,>1} are the successive maxima.

For example, consider the weighing of objects on a

scale missing it's spring. An object is placed on this scale and its weight measured. The 'needle' indicates

the correct value but does not return to zero when the

object is removed. If various objects are placed on the

scale , only the weights greater than the previous ones can be recorded. These recorded weights are the

upper record value sequence. Let Xjj be the highest

water level of a river on the j th day of the i th

location. If one is interested to study at each location

the local maximum values of Xjj, then the local maxima are the upper record values.

Suppose we consider a sequence of products that

may fell imder sets. We are interested to determine

the minimum failure stress of the products

sequentially. We test the first product imtil it fails with stress less than Xj then we record its failure

stress, otherwise we consider the next product. In

general we will record sets Xj^ ofthe m th product if

Xjj^ < min(Xi,...,Xjj^.i), m>l. The recorded failure stresses are the lower record values. One can go fi^om

lower records to upper records by replacing the

original sequence of rv.'s by {Xj, j >1} or if P(Xj > 0) = 1 by { l/Xj, i > 1}. Unless mentioned

otherwise we will call the upper record values as record values. The indices at which the record values occur are given by the record times {U(n)}, n>0,

where U(n) = min{jlj>U(n-l),Xj >Xu(ii.i),n>l} and

U(l) =1. The record times of the sequence {X^ n >!}

are the same as those for the sequence {F(Xjj),>l}.

Since F(X) has an uniform distribution, it follows

that the distribution of U(n) , n > 1 does not depend on F. For a given set of n observations, let Xj ^ < X2 n < ... Xjj^ji be the associated order statistics. Suppose that P{an (Xjj^jj - b^ ) < x} — G{x) a.&n—> 00 . For necessary and suffi- cient conditions about this convergence for various

distributions see Galambos ( 1987). It is well known,

[Ref. [10], that

P(an (W - b„) < x)-±^G(x)|;i:i^^^

s=0

^•

It can be shown that the right side of the above

equation is the distribution fimction of the m th lower

record value. Properties of record values of i.i.d. rvs have been extensively studied in literature, for

example, see Ref [1], Ref [13], and Ref {14], and

1
Ref. [4], for recent reviews. The conditional p.d.f. fj^

of Zu(n) given Zu^jj.n = y can be written as flc(z) = 2f(z)/(l-F(y)).

For many distributions including exponential, Pareto and uniform

E( X u(n) I Xu(n-1) = y) = a + by (1.1)

for some constants a and b . We will say a rv X with

distribution function F belongs to the class C if its n

th record value satisfy the condition (1.1).. In this paper, we will consider the record values of random

variables belonging to class C.

2. MAIN RESULTS

RESULT 1.

If the sequence of rvs Xj, X2, ... , belong to

class C with jQnite variance, then

Cov ( Xu(n) , Xu(ni)) = b^-m Var (Xu(m)), n > m.

Proof:

E( Xu(m+2)) = EE( X U(m+2) I Xu(m+1) = t

)

= EE(a + bt|Xu(ni) = y)

= E ( a + b (a + b y))

= a + ab + b2E(Xu(m))-

In general

E(Xu(n)) = a + ab + ab2+ a b^ + ....+

abn-m-l+ ^n-m.

=(n-m) a+ E(Xu(jn)) , if b = 1.

Thus

CovCXu(m),Xu(n)) = b^ -m Var CXu(m)). if b ;^ 1 = Var(Xu(m)),if b=l.

The following result was proved by Ref. [12].

RESULT 2.

If the sequence of i.i.d. rvs Xj, X2, ... has an

absolutely continuous distribution fimction Fwith

support on [c, d), where c is finite and d may be

infinite and has finite expectation. Further we assume

that F belongs to the class C with b > 0 and with

d = 00 if b > 1 and d = —^if b > 0.

Then 1-F(x) =

b-1

a + (b-l)c

a + (b-l)x^

b-1

for b^l

and 1-F(x) = e'X/a forb= 1,

if and only if E(X u(n)l Xu(n-l) = y) = a + by.

Proof:

Writing the conditional expectation of

^U(n) I Xu(n-1) = y) and simplifying we get

ra 1-

a+by=y+ —Jy i_

d l-F(x)

F(y)

dx (2.1)

Differentiating both sides of the above equation with

respect to y, we obtain

b= (a+(b-l)y)(f(y)/( 1-F(y))) (2.2)

Finally integrating (2.2) with respect to y firom c to x, we obtain the result. For general discussions of result 2 based on

conditional expected values of rv X, see Ref. [7].

It can be shown that for the rv X having the

distribution function as given in the Result 2,

E(X) = a + bcandVar(X)

= (b ( a + ( b - 1 ) c )2) /( 2 - b).

For various results ofthe rv X based record values f

or the case b =1, see Ref [2] and [3]. The results for

the case b >1 are similar to those for the case b < 1

.

2
In this p^r, we will consider the results corre- sponding to b>I. The distribution function cor- responding to b > 1 was introduced by Pickands

(1975) in connection to extreme value distribution. For inference based on record values for the

generalized extreme value distribution see REf [3]. Let fji be the probability density function ofthe n

th record value, Xu(n)- Then

j_b_^a.(b-l)cl b

[b-l a + (b-l)xj b(a + (b-l)cb-l

^^^^^

n! 2b-l

(a + (b-l)x) b-l

(2.3)

It can be shown from (2.3) that

E(Xu(n)) = - l))(b^ ( a+ (b-l) c) -a

)

Var(Xu(n)) =

( b - 1)-2 (a + (b-l) c)2 b^ (( 2- b)-^ - h^).

and Cov(Xu(m) ,Xu(n)) = b^""' Var(Xu(ni)).

We will assume without the loss generality the lower

bound, c of the rv X as zero and ( Y - m ) /s = X,

then

E(Y) = m + asand Var(Y )= a

^ b( 2 - b)"! s 2

For the finite variance, b must be less than 2.

LetTj, T2, ...,Tn be the record values of

Y corresponding to Xu(i),

Xu(2),....,Xu(n).

It can be shown that

n

where Uj, U2 , . ., are independent and identically

distributed with

P(Ui<x) = 1 -x-b/05-1)

Thus • b°-l

E(Tn) = m+ a-——

a

D — 1 and

Var(Tn) =a 2( b - 1)-2 b ^ { (2 -b)-» - b^ } s 2.

Cov(Tn„Tn) = b^-m Var(T^) ^ < „. We can write the Variances and Covariances of T's as Var(Tr) = arbr of and Cov(T^ T^ ) =

arbsC7?.r<s,

where = [ ( 2- b)-r - b^l, b j. = b^ , r= 1,2,

and o- f = a ' (i - 1)"'

There are other distributions see Ref. [5], for which

Cov(Tj.,T5) can be factored out as the product of

two factors, one depends on r and the parameters

and the other depends on s and the parameters.

ESTIMATORS OF m AND s .

The minimum Variance linear unbiased estimator

(MVLUE) of ILI, a of m and s are

^i = Ti-ao

3
Proof.

Let T ' =( Tj, T2, ,Tn), then we can write

E(T) =m L + d C7i

L' = (l,l,...,l),d' = (di,d2,...,dn)and

di = yi-l,i-l,2,...,n.

Let V(T) = o- f I ' s"! = W , W = ( V U )

It can be shown that

yi+l,! ^

yi.i+l ^

b_

(l-b)H b .

i,i_l + 2b-b^|'2-bY

i+l

V ' =

(1-by

Vn,n_ 1 (2-hy

(l-b)H b J

Vij = 0,if|i-j |> 1

Let Wi = a-l((2-b)^)l/2(Tj-b Tj^i), i=l,2,...,n and

To=0.

Then Var(Wi) = s2 and Cov(Wi,Wk) = 0,

i 5t k, 1 < i, k < n.

Suppose W = (Wi,W2,...., W^) and E(W) = Aq,

where q -(m,s)

A'=[AiA2],A; =(di,d2,...,dn),A2 =(ei,e2,...,en),

di = ((2 - b) / bfil - b), Ci = di / (1 - b), i =

2,3,...,n,di = (l/a)((2-b)/b)^^^ ^ and ei = adj

.

Using least squares estimation method, we get on

simplification u = Ti-aa

a = a

-1 f 2-h

'

[ b .

Ti + D

1=2 b I b J

Ti+D-lf2-b

n+1

Var(a) = (4^)a2

b^D

Var(a) = ^ r2-bv fh-i

V b

where

n /2-bV+l

Let a =2 and b = 1.5, then

|i = (17/12) Tj - (1/4) T2 -(1/12) T3 -(1/12) T4

and

a = -(5/8) Ti + (1/8)T2 + (1/24)T3 +(1/24)T4

The corresponding variance and covariance are

Var(a) = —a and

81

Var(a) = —a and

81

Cov(|i,a) = -ya^

4
BEST LINEAR INVARIANCE ESTIMATORS

(BLIE)

The best linear invariant ( in the sense of minimum

mean sqiiared error and invariance with espect to the

location parameter m) estimators

ji, a of |i and a are

-1

and a = a(l + E22)

where [i and a are MVLUE of \i. and a and

( var(ii) cov(ii,a)^ i(^n Ei2^

^cov(|i,a) var(a) )

The mean squared errors ofthese estimators are

MSE(fi) = a(En-E22(l + E22r^)

MSE(^) = a(Eii-Ej2(l + E22r^)

Substituting the values of E j j, Ei2j E22, we get

- . b(b-l)T-2 + b} .

|i = |i -a

. . Db^

a = G

and

MSE =

2 2 a c

b^D

aa

T- {(b-l)T-(2-b)V

MSE(a) = —[(b-l)T-(2-b)]

With n = 4 and b = 1.5, we have

2105„ 37

Ti37 37

1920 640 1920 " 1920

- 1125^ 9 ^ 3 ^ 3 ^

a = Ti + T2 + T3 + T4

1200 ^ 160 ^ 160 160

^

127413 2 MSE(|i) =

10800

MSE(5) = —160

PREDICTOR of Tg

We shall consider the prediction of Tg based on n

observed record values for s > n. Let H* = (hj, h2,...., hj^), where s^ hj =

Cov(Ti,Ts), i = 1,2,. ..,n and gg = s E(Ts -m). It follows from the results of Ref. [9] that the best

linear unbiased predictor (BLUP) of Tg is Tg , where

t, = t+cy,+H'V-'(T-aL-y)

Now

ITV-l = (0,0,...., bs-n) and

ts = A + aYs+b^-"(Tn-a + ayn

=

b^-°

Tn + (1 - b2-^)a + (1 - b^-^)Ysa

The best( unrestricted) least squares predictor of

TsisTs*=Eas |Ti,T2,. ..Jn)-

Thus

^s*^^^ + ^ a + ab^"^(Tn-n)

If we substitute the MVLUE of m and s , then Tg*

becomes Tg.

Let Tg be the best linear invariant predictor of Tg

From the results of Ref. [1 1] it follows that

127413 2 10800

5
where

c*2 = Cov(5, (1 - H' V"^L) A + (y s - H' V"^ 5)ac

and

1 - H' V"h - 1 - b^"^ and y s - H' V"^ 5 = y

Thus

- _ - 2-b

~ — y s-n . 7

Considering the MSE of the predictor, it can be

shown that

MSE(T*) < MSE(ts) < MSE(ts)

REFERENCES

[1] Ahsanullah, M.(1988). Introduction to Record

Statistics. Ginn Press, Needham Heights, MA.

[2] Ahsanullah, M. (1980).Linear prediction of

record values for the two parameter

exponential distribution. Ann. Inst. Stat. Math.

32,A,363-368.

[3] Ahsanullah, M. and Holland, B.(1993) On the use of record values to estimate the Location

and Scale Parameters of the Generalized

Extreme Value Distributions. To appear in Sankhya.

[4] Arnold, B.C., N. Balakrishnan and H.N.

Nagaraja (1992). A fiirst Course in Order

Statistics, John Wiley & Sons, New York.

[5] Balakrishnan,N., M. Ahsanullah and P.S. Chan (1992). Relations for single and product

moments of record values from Gimibel

distribution. Statistics and Probabihty letters, 13,223-227.

[6] Galambos, J.(1987). The Asymptotic Theory

of Extreme Order Statistics, Second edition,

Krieger, Malabar, Florida.

[7] Galambos, J. and S. Kotz (1978) Characterizations ofProbabihty Distributions. Lecture

Notes in Mathematics, 675. Springer -Verlag.

New York.

[8] Goldberger, A.S. (1962). Best linear unbiased

prediction in the generalized linear regression

model. JASA 57, 369-379.

[9] Leadbetter, M.R., G.Lindgreen and H. Rootgen

(1980). Extremes. Springer-Verlag, New York,

N.Y.

[10] Mann,N.R. (1969).Optimum estimators for

linear functions of location and scale

parameters. Ann. Math. Statist. 40,2149-2155.

[11] Nagaraja, H.N.(1970). On a characterization

based on record values. Austra. J. Statist. 19,

70-73.

[12] Nagaraja, H.N. (1988). Record values and

related statistics-A review, Commun.Statist.

Theor.Meth.I7(7), 2223-2238.

[13] Nevzerov,V.B. (1987).Records, Theory of

Probabihty and Its AppUcation 32(2), 219-251.

[14] Pickands, J. ( 1975). Statistical Inference using

extreme order statistics. Ann. Statist.3, 119-

131.

6
Composite Sampling And Extreme Values

Argon, E.D., Gore, S.D., and Patil, G.P.

The Pennsylvania State University, University Park, PA

Issues in environmental sampling and ecological monitoring can involve extreme values

as the main inferential target, or as a design tool for cost-effective sampling. Although

conventional sampling methods address the problem of estimating the population mean

with a desired precision, classical procedures are not always cost-effective for studies

involving extreme values. In this paper, we review some procedures that allow infer- ence on sample extreme values based on sample means while maintaining observational

economy. These procedures use a common sweepout method to identify extremely

large sample values when measurements on composite samples are available. These

procedures' are illustrated with examples in compliance monitoring and enforcement

in hcLzardous waste site characterization. The effect of the compositing design on the

performance of the sweepout method is also investigated. In conclusion, this paper

highlights the need for an investigation of the statistical properties of the sweepout

method.

Keywords. Compliance monitoring. Composite sampling, Concomitants of order statistics, Extreme values, Higher order statistics, Observational economy, Percentiles, Population mean. Ranked set sampling, Site characterization.

1 Introduction

Issues in environmental and ecological monitoring can involve extreme values as a primary

objective for inference or as a design tool for

cost-eflFective sampling. For example, compliance

monitoring and assessment of hazardous waste

sites may require both estimation of the mean

and identification of "hot spots." Choice of the

sampling design must take into account these objectives as well as resource limitations and any

other practical constraints. Although conventional sampling methods address the problem of

estimating the mean with a desired precision,

they may no longer be cost-effective for inference on extreme values. In a different situation, available information on extreme values may be used

advantageously for improving upon the sampling

design. For instance, perceived ranks of sampling units may be used as a sample stratification tool, thereby reducing the required sample

size and/or associated cost while maintaining the

desired precision. In another example, it may be

of interest to estimate fish abundance by sampling from known high-abundance sites.

7
We review some procedures that are based on composite sampling techniques and address the

issue of identifying extreme sample values when

measurements on composite samples are available. In Section 2, we discuss a sweepout method

to identify extreme individual sample values from

composite sample measurements. We illustrate

this method with data on PCB concentrations

in surface soil samples. In Section 3, we discuss

the situation where composites are formed using

two orthogonal contours, and composite samples

are formed along each of the contours. In Section

4, we consider the method of ranked set sampling

as a means of improving the composite sampling

procedure. We evaluate the performance of the

sweepout method when the ranked set sampling

protocol is used to form composite samples. The

Armagh site data is used to illustrate the methods.

1.1 Composite Sampling

A composite sample is formed by mixing several

individual samples or subsamples. The terminology of a "sample" as used here refers to a

physical sample rather than to a statistical sample. For instance, an individual sample is a sin- gle grab collected from the samphng location se- lected for characterization, evaluation, or monitoring. Similarly, a composite sample is a mix of

subsamples drawn from several individual samples. The strength of composite sampUng procedures lies mainly in the physical averaging that occurs due to homogenization of the sample material while forming the composites. Compositing, at least under ideal conditions, incurs no loss of information for estimating population means. However, the loss of information regarding in- dividual sample values, particularly the extreme

values, has been an important limitation of the

method. The available choices have been either to exhaustively measure all individual samples,

or to lose information on extreme individual sample values.

In section 2, we present a statistical method to recover extremely large individual sample values

using composite sample measurements and a few

additional measurements on carefully selected

individual samples. Using available composite

sample measurements, this method first identifies constituent individual samples that may potentially have large values. Obtaining measure- ments on these few individual samples helps re- cover extremely large individual sample values.

The method is illustrated with data on polychlorinated biphenyl (PCB) concentration in surface

soil samples at the Armagh compressor station

along the gas pipeline of the Texas Eastern Gas

Pipeline Company in Pennsylvania (see Ref. [1]). Reference [2] consider this problem in the context

of water quality monitoring, where the maximum

pollutant concentration is either an observed

value or estimated from other measurements.

Noting that the cost of extensive and comprehensive monitoring is prohibitively high, Ref. [2] further note that no method exists that will find

the maximum concentration with certainty unless continuous monitoring is used. They use the

following assumptions in their method:

1. The process of collecting samples is distinct

from their measurement;

2. The cost of sample measurement is high rel- ative to that of collection; and

3. The sample values have, high positive autocorrelation.

The method of Ref. [2] first identifies the composite sample having the highest measured value

and then makes measurements on all the individual samples that form this composite. The

highest observed individual sample value is taken

to be the predicted value of the overall sample

maximum. Assuming that compositing was done

along the time component, this method is based on the premise that in the presence of high positive autocorrelation, the maximum sample value

will tend to appear in the composite with the

highest measurement. The number, and thus the

cost, of tests performed in this method is a constant and is known prior to laboratory analysis.
Under the assumption of no measurement error, Ref. [1] propose an alternative method that is certain to identify the individual sample having

the largest value without measuring all individual samples.

The performance of the sweepout method of Sec- tion 2 is affected by the compositing design. In an extreme case, if composites are formed

with heterogeneous individual samples, then the

sweepout method may perform worse than ex- haustive measurement of all the individual samples. On the other hand, if composites are internally homogeneous, then the sweepout method

can be very cost effective. It then remains to determine how one forms internally homogeneous

composites. Four alternatives are discussed in this paper. First, if a spatial process is known to be operative on the site to be sampled, then locational information on sampling locations may be

used to form reasonably homogeneous composites. We call this situation location-based compositing. Second, if two orthogonal contours are known to exist on the site, then homogeneity

may be achieved by compositing along each of

the contours. In this method, every individual

sample contributes to exactly two composites,

and hence this compositing design is different

from other designs. Third, if information is avail- able on locations with high values, as is common

with fishing activities, sampling may be concentrated only on locations that may yield high val- ues. In this case, all the individual samples are expected to return high values, and hence composites formed from these samples are expected

to be homogeneous. Finally, if sampling units can be compared without exact quantification,

then selected sampling units can be grouped and

ranked, so that composites of sampling units that

are assigned matching ranks will be relatively homogeneous. This rank-based compositing design

is expected to enhance the performance of the

sweepout method of Section 2. Each of the composite designs is illustrated with data on PCB in surface soil at the Armagh Compressor Station of

the Texas Eastern Gas Pipeline Company. (See

Ref. [3,4]). <

1.2 The Armagh Site

Location and Features. The Armagh compressor station is located in West Wheatfield

Township, Indiana County, PA. The site includes one compressor building along witli several other

buildings on 79 acres. There are two known liquid pits. There is one wetland situated within

one-half mile of the site. Richard Run, which

flows to the south of the site, is classified as a

cold water fishery. There are no public recre- ational facilities near the station. Onsite soils are defined as being within the confines of the sta- tion site fencing and are accessible only to Texas

Eastern personnel and authorized site visitors.

Onsite Surface Soil Sampling. Potential

sources of PCB had been identified and a rectan- gular grid was laid out around each such source. Four different onsite grids were identified by the

alphabetic codes "A" through "D". Grid points were identified by a two-digit row number and

an alphabetic column code. Sampling of the sur- face soil was done at selected grid points in two

distinct phases. Grid "D" was not sampled during Phase I, and as such is not included in the

illustration here.

The distance between consecutive rows as well

as between consecutive columns was 25 feet. Soil

samples were taken from a 0-inch to 6-inch depth.

After removing vegetation, rocks, and other debris, the sample at each grid point was thoroughly mixed to obtain a homogeneous sample

for analysis and quantification.

2 Sweepout Method to Identify Extreme Sample Values

Let xi,X2., - ^ denote the individual sample

values and let y be the composite sample measurement. Further, let X(^k) denote the maximum

of the k individual sample values. That is,

X(fc) = mdix{xi,X2, . . . ,Xk}.

Observe that

y < X(k) < ky.
This inequality implies that the measurement on

a composite sample gives bounds for the largest

constituent individual sample value.

Now consider two composite samples of sizes k-i and k2, with measurements y-i and y2-, and having the largest individual sample values X(^j^^) and

X(k2)^ respectively. Without loss of generality,

suppose that t/i < 2/2- general, this does

not allow for comparison between Xf^).^-^ and X(^).^y However, if /^it/i < 2/2, then X(^j) < ^(/j^) ^^id

hence it is no longer necessary to consider the

first composite sample when the individual sample having the largest value is of interest. In this

way, a number of composites can be eUminated

without any additional testing. This ehmination

process leads us to fewer composites which may

possibly contain individual samples that have

large values. Measurements on individual samples in these composites then help identify the

individual sample having the largest value.

Using this reasoning for identifying the largest

individual measurement we obtain the sweepout

method of Ref. [5] as follows:

1. Identify the composite sample having the

largest measurement, say y^ax^ s^nd of size

2. Measure all the individual samples in this

composite and identify the largest individual

measurement, say Xmax3. Consider the next largest composite measurement, say y*, on a composite sample

of size k*. If Xjnax > f^*y* then Xmax is the largest individual measurement and the

search is stopped.

4. If Xjnax ^ measure every individual

sample in this composite and identify its largest individual measurement, say x*

.

5. If Xmax < X* then x^ax is redefined and as- signed this new largest value x* . Repeat

from (3) until the largest individual measurement is identified. 1

Reference [5] illustrate the sweepout procedure

with an application to simulated composite sample values of PCB concentrations in surface soil samples at the Armagh Compressor Station. Table 1 shows the individual sample values and simulated composite sample measurements. In this

illustration, Gore and Patil found that only 8

additional measurements on individual samples were required to identify the largest individual

sample value from among a total of 358 individual samples. There were 90 measurements al- ready made on composite samples.

Figure 1 shows a scatterplot of individual sample values plotted against the simulated composite sample measurements. The two rays through

the origin indicate the bounds on the largest

individual sample values. Since 4897.5 ppm is the largest composite sample measurement (composite number 25 in Table 1), constituent individual samples of this composite are measured

separately. This identifies an individual sample with a PCB concentration of 10000 ppm. A

horizontal line at the height of 10000 ppm in- dicates that there is only one more composite

(composite number 5 in Table 1) which can possibly contain an individual sample with a PCB

concentration of more than 10000 ppm. Making

measurements on all the individual samples constitutiiig this composite identifies an individual

sample with a PCB concentration of 10700 ppm.

There is no other composite that can contain an

individual sample with a PCB concentration ex- ceeding 10700 ppm, as is evident from Figure 1 (b). Thus, making measurements on 8 individual

samples constituting two composites has identified the individual sample with the largest PCB

concentration.

2.1 Extensive Search of Extreme Values

The sweepout method described above can be ex- amined further for its cost effectiveness in identifying extreme values. Note that exhaustive test- ing of all individual samples (without compositing) identifies all individual values. In this case.
Table 1: Individual sample values and simulated composite sample measurements

,

Composite Individual Composite Composite Individual Composite

Sample Sample Sample Sample Sample Sample

Values Measurement Values Measurement

01 2.9, 3.1, 22, 22 12.5 46 1.9, 1.6, 82, 390 118.9 02 21, 298, 18, 1880 554.3 47 1.4, 1, 530, 320 213.1 03 9.4, 51, 319, 1.0 95.1 48 160, 180, 19, 320 169.8 04 105, 30, 22, 67 56.0 49 5.4, 1.7, 0.0, 15 5.5 05 18, 2320, 10700, 2960 3999.5 50 7.7, 6.9, 310, 19 85.9 06 38, 2.5, 13, 154 51.9 51 27, 23, 21, 5 19 07 1.1, 12, 55, 8.7 19.2 52 7.5, 2.2, 55, 80 36.2 08 13, 1.9, 2.9, 22 10.0 53 7.7, 4.3, 24, 250 71.5 09 129, 12, 44, 22 51.8 54 4.3, 6.4, 20, 33 15.9 10 1.6, 1070, 1.0, 64 284.2 55 436, 9.5, 120. 21, 58 128.9

11 13, 3.8, 3, 6.8 6.9 56 1.5, 160, 180, 1000 335.4 12 13, 3.8, 2.8, 6.9 6.1 57 2.9, 15, 150, 12, 11 38.2

13 34, 28, 745, 3850 1164.3 58 2.9, 26, 1.2, 1.3 7.9 14 50, 18, 17, 34 29.8 59 24, 2.6, 3.5, 18 12.0 15 4.6, 22, 1.0, 42 17.4 60 3.9, 27, 5.4, 12 12.1 16 14, 3.3, 1.5, 2.6 5.4 61 72, 38, 7.1, 35 38.0 17 2.4, 1390, 3, 672 516.9 62 52, 37, 66, 38 48.3

18 8.9, 661, 20, 18 177.0 63 1.3, 2.1, 15, 4.4 5.7 19 18, 24, 26 22.7 64 60, 79, 8.7, 150 74.4 20 3.5, 16, 20 13.2 65 16, 24, 18, 160 54.5

21 97, 70, 14, 150 82.8 66 150, 210, 18, 13 97.8 22 37, 72, 40, 33 45.5 67 26, 7.8, 43, 49 31.5 23 38, 44, 83, 30 48.8 68 46, 24, 18, 12 25 24 38, 100, 140, 47 81.3 69 38, 12, 140, 60 62.5 25 590, 7100, 10000, 1900 4897.5 70 26, 14, 190, 61,33 64.8 26 670, 940, 240, 290 535 71 340, 190, 10 180

27 74, 200, 120, 220 153.5 72 0.0, 0.0, 0.0, 0.0 0.0 28 280, 260, 10, 250 200 73 0.0, 0.0, 0.0, 1.1 0.3 29 44, 110, 660, 230 261 74 1.1, 2.8, 4.2, 6.6 3.7 30 580, 1100, 1300, 4900 1970 75 6.9, 16, 7, 13 10.8 31 110, 80, 210, 12 103 76 11, 13, 6.4, 8 9.6 32 75, 890, 170, 550 421.3 77 0, 236, 7.2, 2.4 61.4 33 2300, 420, 520, 1300 1135 78 5.8, 535, 1.1, 0.0 135.5 34 0.0, 1.2, 1.67 2.5 79 0.0, 1.4, 4.9, 0.0 1.6 35 5.7, 17, 4.3, 36 15.8 80 0.0, 0.0, 5.1, 6.3 2.9 36 28, 170, 10, 62 67.5 81 7.9, 14, 20, 31 18.2 37 300, 6.4, 53 119.8 82 52, 1, 500, 46 162.3 38 16, 18, 150, 27 52.8 83 16, 5, 36, 64 30.3 39 6.2, 7.1, 31, 38 20,6 84 40, 38, 68, 7.5 38.4 40 16, 66, 61, 340, 1500 396.6 85 40, 33, 36, 17 31.5

41 1.3, 3.5, 2.1, 8.8 3.9 86 35, 4, 170 52.3 42 7.5, 2.7, 1.6, 11 5.7 87 110, 200, 4.2 104.7 43 0.0, 0.0, 17, 2.8 5.0 88 7.4, 3.3, 21, 2.3 8.5 44 1.1, 5.9, 350, 17 93.5 89 3,8, 35, 20, 17 19.0 45 3.2, 5, 11, 5.1 6.1 90 23, 17, 3, 6,8 12,5

11
Composite Sample Values Composite Sample Values

Figure 1: Illustration of the sweepout method. Individual sample values {Y axis) vs composite sample

measurements {X axis) in thousand ppm. (a) The upper and lower bounds for the largest individual

values, (b) Measurements on individual samples from only two composites identify two largest individual values; (c) Measurements on individual samples from only two composites identify the three

largest individual values, (d) Measurements on individual samples from three composites identify the

four largest individual values.

12
identification of extreme values is achieved simply by arranging the individual values in a descending order of magnitude. Thus, the method

of exhaustive testing involves as many measure- ments as the number of individual samples. For

example, the case of the Armagh site would re- quire 358 measurements.

In order to investigate the relationship between

the number of extreme values identified and the

number of measurements made, we extend the

sweepout method to all the 90 composites at the Armagh site. Figure 2 gives a graphical

summary of these results. The concavity of the curve implies that identification of every additional extreme value initially requires relatively more measurements.

As a consequence of its ability to identify individual samples having large values, the sweepout

method can also provide estimates of upper percentiles of the distribution of individual sample

values. Reference [.5] discuss the applicability of

this feature of the sweepout method to compli- ance monitoring and to quality assurance management. See Ref. [6] and [7] for more details.

3 Two-dimensional Compositing Design

The Sweepout method discussed above assumed

that every individual sample contributes to ex- actly one composite sample. If each individual

sample is allowed to contribute to more than one

composite, it is possible to expedite the search

for extreme individual sample values using composite sample measurements. For instance, ar- ranging the individual samples in a rectangular

array allows us to form composites by combining

all the individual samples in each row to form

row-composites, and all the individual samples

in each column to form column-composites. In

this way, a row- column arrangement implies that

every individual sample contributes to exactly

two composites, a row-composite and a columncomposite.

For example, consider the situation where 64

individual samples are available, and the problem is to identify the largest individual sample

value using a minimum number of tests. One

possible method is to form 16 composites, each

of size 4, make 16 measurements on the composite samples, and then apply the sweepout

method. Four additional measurements will be

made for every composite sample selected for

retesting. According to the row-column arrangement described above, these 64 individual samples could be arranged into a square having 8 rows and 8 column?. This arrangement will

produce 8 row-composites each of size 8 and 8

column-composites each of size 8. Thus there

will be 16 measurements on the 16 composite

samples. Suppose {Xij, i = 1,...,8; j —

1, . . . ,8} denote the 64 individual sample values;

{Yi.,i = 1,...,8} denote the 8 row-composite

measurements; and = 1, ... ,8} denote the

8 column-composite measurements. Suppose for some i* , Yi*. is the largest row-composite measurement, and for some j*, Y.j* is the largest

column-composite measurement. Then the individual sample in the' i*th row and the j*tl\ col- umn is likely to have the largest value. Therefore, the value of Xi*j* is determined by making a measurement on this individual sample. Having

known the value oi Xi*j*, it is then possible to determine whether any other individual sample is likely to exceed Xi*j* by comparing the measure- ments on the other row-composites and columncomposites with Xj*j+/8- Note that every retest- ing stage now involves only one additional measurement, as opposed to four additional measure- ments in the case of the linear sweepout method

of the preceding section.

We illustrate this procedure with an application

to the Armagh site. For this purpose, we consider only part of the data given in Table 1, so that we have a square of 8 rows and 8 columns.

Thus, selecting only the grid points from Grid A

between rows 50 and 57, columns 8 and 15, we

form 8 composites.

As can be observed from Table 2, the row- composite of row 54 has the largest measure- ment, while among the column-composites, col-
0 <0 eo 120 1«0 200 240 280 320 360

Figure 2: Number of composites retested {Y axis) vs number of extreme values identified {X axis).

The diagonal line represents the optimal case in which exactly 4 extreme values are identified for every

composite.

Table 2: Row and column arrangement of sampling units for compositing. Here, every individual

sampling unit contributes to exactly two composites.

Column

Row 8 9 10 11 12 13 14 15 y.. 50 3.0 2.9 3.1 1.9 1.6 1.4 1.0 2.129 14.9 . 51 16 22 22 82 530 160 138.7 832

52 20 21 298 9.4 390 320 19 320 174.7 1397

53 18 18 1880 51 319 105 18 2320 591 4729

54 24 34 1 30 10700 2960 2291 13749

55 26 28 745 3850 22 38 2 673 4711

56 50 18 11 67 13 154 89 313

57 17 34 3.8 1.1 12 55 20.48 122.9 21.75 20.56 497 501 237 134.6 1543 785

k] Y.J 174 144 2982 4010 711 1077 10801 5971

14
umn 14 gives the largest measurement. The proposed sweepout method suggests that the individual sample in row 54 and column 14 be measured. This gives a value of 10700 for this particular individual sample. It is then compared with

the upper bounds formed from the remaining

composite measurements. Since there is no other

composite sample with an upper bound that ex- ceeds this value, we conclude that 10700 is indeed

the largest individual sample value among the 55

individual samples involved in the above illustra- tion. Note that it took only one measurement

on an individual sample to identify the largest

individual sample value. Thus, the total number of measurements required for identification

of the largest individual sample in this example

is 17, with 16 measurements for the 16 composite

samples and one measurement for the individual

sample in row 54 and column 14.

4 Compositing a Ranked Set

Sample

For predetermined positive integers m and r, ranked set sampling (RSS) involves selection and

acquisition of vn?r units, of which only mr units are quantified. First, m random samples, each of

size m, are randomly selected from a population

(with distribution function F^ mean variance

(7^, say). The m selected units in each sample

are ranked by a judgement process such as vi- sual inspection or any other inexpensive method

which does not require actual measurement. The

unit with the smallest rank is quantified from the

first sample, the unit having the second smallest

rank is quantified from the second sample, and so

on, until the unit with the highest rank is quantified from the m-th sample. Thus, m units are quantified out of the rn?' units originally selected.

The process is repeated r times, thereby providing a total of mr measurements which constitute

the ranked set sample. Reference [8] and [9] provide mathematical formulation for this sampling

method which was introduced earlier by Ref. [10] as an improvement in simple random sampling

(SRS) for estimation of mean pasture and for- age yields. This method is particularly attractive

when quantification of units is difficult or expensive, but ranking of a small set of units can be

done with a reasonable accuracy even without

making measurements.

RSS may be utilized advantageously for forming

internally homogeneous composites as compared

to those based on random groupings. With m

samples of size m, we can form m composites by

physically mixing sampling units on the basis of

their ranks. Likewise, we get mr composite samples of size m from rrt^r units. These samples, in

turn, provide mr measurements. The standard

deviation of these measurements is expected to be smaller than that of the same number of measurements obtained from composites comprising

sampling units selected randomly, m at a time,

out of rri^r available units in most cases. For ex- ample, in the case of 64 sampling units, 16 sets of size 4 are formed for the purpose of ranking.

These 16 sets are tabulated in Table 3. The graph in Figure 3 shows the number of measurements (on the Y axis) versus the fiumber of

extreme values identified (on the X axis). Here,

the data set of Table 2 is used with three differ- ent compositing designs. First, the 64 sampling

units were grouped in 16 sets of size 4 each based on the contiguity of their locations. Thus, every

2x2 square within the 8x8 arrangement of the 64

sampling units is used to identify the individual

sampling units for compositing. We call this design "contiguity-based" compositing. Next, the

8 rows were used to form 8 row-composites, and

similarly for 8 columns. We call this design "rowcolumn" compositing. Finally, forming 16 sets of

size 4 each, we rank the 4 sampling units within

each set, then form 16 composites from sampling

units that were assigned the same ranks. In other

words, the compositing is based on contiguity of

ranks, rather than locational contiguity, as in the

first case. We call this last design "rank-based"

compositing. It is easily seen that the rank-based

compositing has performed better than the other

two compositing designs. The difference between

the three graphs is due only to the compositing
5 5-'

T

' • ' ' 1 ' 1

1 1 1 1 1 11 i 1 i i 1

1 . 1 11 1 1 1 1 H

' ;c i& je ?7 4$ 5 5 Figure 3: Number of measurements vs. Number of extreme values identified, solid line: row-column

compositing; dotted line: contiguity-based compositing; and broken lines: rank-based compositing

design, since the same 64 sampling units are used

in the illustration. For additional references, see Ref. [11-21].

5 In Conclusion

This paper highlights the need for an investigation of the statistical properties of the sweepout method. The performance of the sweepout

method will be determined by several factors, in- cluding the autocorrelation structure among the

individual sample values, the statistical distribution of these individual sample values, and the

compositing plan as well as the composite sample size.

6 Acknowledgements

This paper has been prepared with partial support from the United States Environmental Protection Agency Grant Number CR-821531. The

contents have not been subjected to Agency review and therefore do not necessarily reflect the

views or policies of the Agency and no official endorsement should be inferred.

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Studies on the applications of composite

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surface soil sampling for PCB at the Armagh Site, Technical Report No. 92-0305,

Center for Statistical Ecology and Environmental Statistics, Department of Statistics,

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Park, PA. 16802, 1992.

[2] Casey, D., Nemetz, P. N., and Uyeno, D.,

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and Assessment, 5 (1985), 165-176.
Table 3: Sets of sampling units for the purpose

of ranking. Sampling units that are assigned

the same rank are then composited together to achieve internal homogeneity of composite samples.

Set Unit 1 Unit 2 Unit 3 Unit 4

1 2.9 18 24 17

2 3 18 26 18

3 16 20 28 34

4 22 21 * 50

5 1.9 9.4 1.0 3.8

6 3.1 51 34 11

7 22 298 745 *

8 82 1880 3850 *

9 1.4 105 22 1.1 10 1.6 319 30 67

11 530 320 *

12 * 390 *

13 1 18 2 12

14 160 19 38 13

15 * 320 2960 55

16 * 2320 10700 154

*: no sample collected from this location

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300.

17
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G., and Smith, D. W., Evaluation of ranked

set sampling for estimating shrub phytomass in Appalachian oak forests, Publication Number FWS-4-80, School of Forestry

and Wildlife Resources, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1980.

[16] Patil, G. P., Sinha, A. K., and Taillie,

C., Ranked set sampling. In: Handbook of

Statistics Volume 12: Environmental Statistics (eds.: G. P. Patil and C. R. Rao), North

Holland/Elsevier Science Publishers, New

York, Amsterdam, 1994, pp. 167-200.

[17] Patil, G. P. and TailUe, C., Performance of

the largest order statistics relative to the

sample mean for the purpose of estimating a population mean, Bull. Int. Statist. Inst., 54 (1991).

[18] Patil, G. P. and Taillie, C., Environmental

sampling, observational economy, and statistical inference with emphasis on ranked

set sampling, encounter sampling, and composite sampling. Bulletin of the International Statistical Institute, Proceedings of

the 49th Session, (1993), 295-312.

[19] Sarhan, A. E. and Greenberg, B. G., Contributions to Order Statistics, Wiley, New

York, 1962.

[20] Stokes, S. L. and Sager, T., Characterization of a ranked set sample with application

to estimating distribution functions, Journal of Applied Probability, 83 (1988), 374-

381.

[21] Watterson, G. A., Linear estimation in

censored samples from multivariate normal

populations. Annals of Mathematical Statistics, 30 (1959), 814-824.

18
Extremal Sojourn Times For Markov Chains

Arnold, B. C.

University of California, Riverside, CA

Consider a continuous time Markov chain with state space {1,2,3,...}. Since sojourns in

particular states are independent exponential random variables, it is possible to derive the

asymptotic distribution of the maximal and minimal sojourn in a particular state or in any

state. Discrete time analogies axe described and the more challenging problem of deriving

the distribution of the extreme sojourn times in a particular group of states in discrete time

is introduced.

1 Extremal sojourns

Consider X{t) a continuous time

Markov chain with state space {1,2,...}.

Assume that the chain is irreducible and

ergodic with intensity matrix Q satisfying

Ql = 0 (1)

and denote the long run distribution by tt. For a particular state i, sojourns in that

state are i.i.d. exponential {—qu) random

variables. During a time interval (0,T], a

particular state i wiU be visited a random

number of times.

If we let Ni{T) denote the number of visits

to state i, it is easy to verify (using results

for delayed recurrent events) that

Ni{T)/T

^

{-qu)7ri . (2)

For more detail see Ref. [1] where the finite

state space case is discussed.

E we use Xj^^ to denote the jth. sojourn

in state i, then our interest is in studying the asymptotic behavior of extremal so- journ times defined as follows.

Mi(T) = max X)

m{(T) = min X]

j<N,(T) ^(0

19
M(r) = max M,(r)

and

m(T) = min m,(T) . t

To determine the asymptotic distribution

of these random variables, we need a re- sult of BarndorfF-Neilson (Ref. [2]) (which

may be conveniently found as part of Theorem 6.2.1 in Ref. [3]). It states that

if Xi,X2t... are i.i.d. in the domain of

maximal attraction of some distribution A

and if N{T)/T —> 6 then max_,<;v(T) -^j

has the same asymptotic distribution (with

the same normaHzing constants) as does

maiXj<sT ^j- We also need the observation

made in Ref [1] regarding the maximum

of heterogenous exponential random variables. The result in question deals with

two independent sequences Xi,X2,.-- and

Yi,Y2,.... The X,'s are exponential Aj)

and the l^'s are exponential(A2) where Aj <

A2. For Qre(0, 1) we have

max[max X,-, max Yi\^mdixXi

i<an t<(l—Q)n i<an

where ~ denotes "hcis the same asymptotic

distribution". Finally we recall that the

minimum of m arbitrary exponential ran- dom variables is again exponential with a parameter obtained by adding the parameters of the individual exponential variables.

These observations allow us to state the

following results for the maximal and minimal sojourn in a particular state. Theorem 1: For any state i

lim P{{-qu)M,{T)-log{-qiiTii)-logT < z)

T—*oo = exp(— e~^), 2 e R

and

lim P((-9„f7riTm.(T) < z) = 1 - 6"%

T-*oo

z > 0 .

To obtain limiting distributions for ex- tremal sojourns in any state, some minor

regularity conditions must be imposed. We

assume there exists a smallest {—qjj) and

reorder the states so that —qn < —922 ^ — With this convention we find

Theorem 2: (A) If for some integer k

and some e > 0 we have

(-?ii) = (-922)'= ••• = i-qkk) < {-qjj)-e,

then

lim P((-9ii)M(r)-W(-9nE7r,)-logr <

= exp(— e~^), 2 e R .

(B) If the series II-^i(— 9u)^7r,- is convergent, then

00 lim P(E(-g.,)'^.]rm(r) <z) = l-e-^

z>0 .

2 Matching chains

A more interesting problem involves

the study of two independent Markov processes X{t)^ y(^) and identifying the largest

20
time interval during which X{t) = Y{t).

In continuous time this is a trivial extension of the material in the Icist section. We

can identify {{11,12) '• h = 1,2, ...,Z2 =

1,2,...} as the state space of the process

{X{t),Y{t)} and we are interested first in

the maximal exponential sojourn in the

state (f , i) for each i and then in the maximum of these maxima. As we shall re- mark in the next section, the study of

long matches between discrete time Markov

chains is considerably more complicated.

3 Discrete time

If we study a discrete time Markov

chain with state space {1,2,3,. . .} then so- journs in a particular state are i.i.d. geometric random variables. Maxima of i.i.d. geometric random variables cannot be nor- malized to converge in distribution (see e.g. Ref. [4]). However we can get useful approximations by using the following observation. If X is geometric(p) (i.e. P{X =

k) = p{l - p)''-\ k = 1,2,...) then we

may introduce an exponential (— log(l —p))

random variable whose integer part is identicaUy distributed with X and consequently

we have

W < X <W + 1 and we can bound maxima of X's by maxima of Ws. Ref. [1] provides details in the

finite state space case. Only minor regularity conditions (analogous to those in Theorem 2 above) are required to extend the

results to the infinite state space case.

Now, what about long matches between

two independent discrete time Markov

chains? Again we may combine the two

chains Xn and Yn to form {Xn,Yn) with

state space {1, 2, . . .} x {1,2, . . .}. However,

now a match occurs when the two dimensional chain remains in the class of states

(1,1), (2, 2), Sojourns in such classes of

states have more comphcated distributions

than sojourns in particular states (which

are geometrically distributed). Exceptions occur if the chain is "lumpable" without

upsetting the Markov property, but most

interesting examples do not have this property.

REFERENCES

[1] Arnold, B.C. and Villasenor, J.A. "The

distribution of the maximal time to departure from a state in a Markov chain",

in Statistical Extremes and Applications, J. Tiago de Oliveira (ed). Reidel, Dordrecht,

1984, pp 413-426.

[2] Barndorff-Nielsen, 0. "On the limit distribution of the maximum of a random

number of independent random variables".

Acta. Math. Acad. Sci. Hungar. 15, 1964,

399-403.

[3] Galambos, J. The asymptotic theory of

extreme order statistics. Wiley, New York,

1978.

[4] Anderson, C.W. "Extreme value theory for a class of discrete distributions with

appHcations to some stochastic processes".

Journal of Applied Probability 7, 1970, 99-

113.

21

Bootstrapping Extremes Of I.I.D. Random

Variables

Athreya, K.B. and Fukuchi, J.

Iowa State University, Ames, lA

Let Xi,X2, • • • be i.i.d. random variables with common distribution function F. Define

Xn:n = max(Xi , X2, . • • Xn). Assume that there exist > 0,6^ € lR.,n > 1 such that

Gn{^) = Pia-niXn-.n " < ^} couverges to one of Gnedenko's extreme value distributions. In this paper the problem of estimating Gn{x) by the bootstrap technique is considered.

We define different bootstrap distributions for different types of domain of attraction that

F belongs to. It is shown that both when a„ and bn are known and when and 6„ are estimated from the data the bootstrap distribution is weakly consistent if m=o(n) and it

is strongly consistent if m==o(j^^). These results are applied to the problem of obtaining

confidence intervals for the upper end point of the support of F.

1 Introduction

Since Efron (Ref. [1]) introduced the bootstrap

method of approximating sampling distributions

of statistics, many papers have investigated its tisymptotic properties. One of the desired properties of this method is the consistency, namely,

the Umit of the bootstrap distribution is the same

as that of the distribution of the original statistic. Examples of the situations where Efron's bootstrap (the simple random sampling from the original data) fail to be consistent are, among others,

the sample mean of heavy tailed random variables

(Ref. [2], [3]), the sample mean of weakly dependent random variables (Ref. [4], [5]), normalized

maximum of i.i.d. random variables (Ref. [6]). We study asymptotic properties of bootstrap for the distribution of normalized extremes when the

underlying distribution belongs to the domain of

attraction of an extreme value distribution.

'Research supported in part by NSF Grant DMS 92- 04938, 1991 Mathematics subject classification 62G05 62G30

Keywords and phrases: extremes, bootstrap.

In Section 2, we investigate the inconsistency,

weak consistency and strong consistency of bootstrappng a„(Xn:n — with appropriate choice of

resample size m=m(n) when and bn are known.

In Section 3, the same problem as in Section 2 are

investigated when and bn are unknown. In Ref.

[6], it was pointed out that the naive bootstrap of

the maximum of uniform i.i.d. random variables

with m=n fails to be consistent. In Ref. [7], it was shown that when F belongs to the domain of attraction (in the sense of extremes) of one of the extreme

value distributions, the bootstrap distribution of

maximum converges to a random distribution. Recently, Deheuvels, Mason and Shorack (Ref. [8])

proved the weak consistency and the strong consistency of bootstrap for the normalized maximum

when normalizing constants are estimated. They

utilized von Mises's parameterization of extreme

distribution and thus their method does not need

the knowledge about which type of domain F belongs to. Results in this paper were obtained independently of Ref. [8] and our approach is different

23
from theirs. We define different bootstrap distri- butions for different types of domain of attraction, so in practice we need to know which type of domain F belongs to. But our version of the bootstrap distribution is more appropriate when infer- ences for population parameters such as obtaining

confidence intervals are concerned.

2 Limits of bootstrap distributions : when normalizing constants are known

We begin with a review of basic results from ex- treme value theory. Let Xi, X2, • • • he i.i.d. random variables with a common distribution function F and Xi:n < ^2:n < *

' * < -^n:n be the Cor- responding order statistics. Let F~^{u) := inf{x : F{x) > u} be the left continuous inverse of F and

Cg be the set of continuity points of a function

G. We say that F G D(G) if there exist constants

a„ > 0,6n € R and nondegenerate distribution

function G such that

P{aZ\Xn..n - bn) < x} ^ G(x) ^ Cg- (1)

It is known (cf. Ref. [9]) that G must be one of the

following types, (up to location and scale changes)

G{x) = A(x) = exp(-e~^) x G R,

G{x) = ^a{x) =

0 x<Q

G{x)

QX^{-X~°') X > 0,

exp(-(-a:)'") x < 0

X > 0,

where o: > 0. In (1), an,bn can be chosen as follows: (cf. Ref.

[9])

FeD{A): an = F-\l - —) - -Tn, bn = in, en F 6 D{^a) : an = 7n, 6n = 0 , FeD{^a)- an = dp -In, bn = Op,

where 7^ := F~^{\ - ^) and Op = sup{x : F{x) < 1}.

Let (fi,.F, P) be a probability space and let Xi,X2,---, be a sequence of i.i.d. random variables on with a distribution F€ D{G) where

G=A or $Q or Let m = m{n) G N be

such that m(n) —^ 00 cis n ^ 00. Given X„ := {Xi , X2, • • •

, Xn), let Yi,Y2, - Ym be conditionally

i.i.d. random variables with the distribution

1 P{Yi=X,\Xn) = -, J = 1,2, n,

i.e. {Yi,Y2, • •

' ,Ym) is a i.i.d. resample of size m

from the empirical distribution of Xn- Let Yi.,m <

y2:m < • • • < Ym:m be the Corresponding order

statistics. Now, define

Gn{x) = Pia-\Xn..n-bn)<x),

Hn,mix,Oj) = P(a~\y^:^ - 6^) < x|Xn),

foT u> E ^ and call Hn,Tn{x,(^) the bootstrap distri- bution of a~^{Xn:n - &n)- The first subscript n of H

represents the original sample size and the second

subscript m of H represents the resample size. The

next theorem shows that if m=n, Hn,m{x,(^) has a random limit and thus the naive bootstrap fails to approximate Gn{x). Let PRM(//) denote a Poisson

random measure with mean measure //(•). Theorem 1 Let F G D{G) where G is an extreme

value distribution and let an > 0, bn n > 1 be such

that (1) holds, (i) If G = A, then for any x,- G R,

i = l,---,r,

where

{Hn,n{Xi,Uj),i= 1,2, •••,r)

{H{xi,oj),i= l,2,---,r),

H{x,uj) := exp(-r((x,oo),a;)),

(2)

andT{-,uj) is a PRM(fi) on ;B((oo, oo]) f5((oo,oo])

denotes the Borel a-algebra on (oo,oo].j with

H{B)= ( e-^dx V5 G ^((oo,oo]). (3) Jb

(ii) If G = ^a, then for any x,- > 0, i = 1, • • •, r,

{Hn,n{xi,uj),i = l,2,---,r)

{H{xi,u}),i= l,2,---,r). (4)

24
where

JT(x,a;):=exp(-r((x,(X)),u;)),

andT{-,(jj) is a PRM(^) on 5((0, oo]) with

fi{B)= f

x-^'dx VjB G>B((0,oo]). (5) Jb

(Hi) IfG = ^Q, then for any Xi < 0, i = I, - • ,r,

{Hn,n{xi,u;),i= 1,2, •••,r)

{H{xi,uj),i=l,2,---,r), (6)

where

H{x,u):= exp(-r((a:,0],a;)),

and T{-,uj) is a PRM(^) on B{{—qo,0]) with

fi[B) = I {-xY dx V5 G ^((-00,0]). (7) Jb

Proof. We can write

= F::{anx+bn)

n{l - Fr^janX + bn))

y

For (i), define a point process Tn on (—00,00] by

n

where €a is the delta measure at a. Then, by

corollary 4.19 of Ref. [10], Tn converges weakly

to a PRM{n) where /i is given by (3). Therefore

the continuous mapping theorem gives the result. Proofs for (ii) and (iii) are similar.

We note that the condition m = n is not neces- sary for the above results to hold. Even if m/n —* c, 0 < c < 00, results similar to Theorem 1 hold.

Theorem 1 can be easily extended to the bootstrap for the joint distribution of a~^{Xn:n — ^n),

a-l(Xn_l:n " &n), • ' " , a-^(Xn-r+l:n " K)- NoW defineK := {{ki, • • •

, kr-i) : fci > 0, i = 1, • • • , r - 1,

kii- k2-\- • + kj <jj= l,2, = --,r- 1},

and for xi > X2 > - • • > Xr,

(logG(xi)-logG'(a:2))'^^

(ki,-kr-l)eK

(log(;(a:,-i)-log6'(a:.))^-^

G{Xr).

It can be shown that (cf. Ref. [11] for the case

r=:2) if (1) holds for some nondegenerate distribution function G, then

P{a~'^{Xn:n-K) < Xi,a~'^{Xn-l:n-bn) < ^2,- "

^ T't{x\ ,

• • •

, Xj-^.

However, we have

Theorem 2 Suppose that (1) holds. Then

P{a~'^{Yn:n " &n) < Xi, a~^{Yn-\:n - bn) < X2, • •

,

E

(A:i,-,fcr)eK

(T(x2,a;)-r(a:i,a;tf

ki\

kr-l\

where T(-,u;) ^5 a Poisson random measure given

in Theorem 1 and T(x,u)) = T{{x,oo),uj) by defi- nition.

But suitable choices of m make the bootstrap consistent.

Theorem 3 Suppose that (1) holds. If m(n)=o(n).

sup I ^n,m(n)(a;,'^) " G{x) \^ 0, (8)

in probability. Moreover, ifYl^=\ < oo, VO < A < 1, then (8) holds w.p.l . (w.p.l stands

for "with probability 1".)

25
Proof. We will write m=m(n) for convinience. Let

1 Fn[x) := — ^

/(_oo be the empirical distribution of Xi , X2 , • • , Xn. Then

{

and

f mil-Fr,{amX-\-bm))

m y/

'

771 m{l - Fn{amX + hm)) = —n{l - Fn{amX + bm)) nm

= —Tn,m {say), n

Let := 1 - F{amX + bm). Then (1) implies that

mpm c{x) = -logG(x). Thus

771 E{—Tn,m) - 'rnpm c(x), n

TTl TTl VaT{ —Trt,m) = —mp^(l -pm)-^0-

n n

Therefore

777(1 - Fn{amX + 6^)) c{x)

in probability Vx G R. Hence

in probability Vx G R.

Since ^Tn,m = ^(^n.m - J^Pm ) + "^Pm and

J^Pm — c{x), we need show that ^{Tn,m - npm)

0 w.p.l to prove that ^Tn,m c{x) w.p.l . By the

Borel Cantelli lemma, it is enough to show that

n=l

J2 P{\ '^('^n,m - npm) |> f) < OO Vf > 0.

Let (fmi^) = Pm€^ + (1 - Pm) be the moment generating function of Bernoulli {pm) distribution.

Then > 0,

771 , ^ . 777 _ — log P{ —{Tn,m -Tipm)> i) n n = -logP(e^(^"--"P-) >e^^^) (9)

77 < -loge-^^^^re^^^"--"?-))

n = -e€~0mpm-{- log ^m{Or

-^€-^c(x) + log(e'^(^)("'-^)) = -$€ + c{x){e^ -1-6)

= f{e,e) (say).

By taking the derivative of f{6, f), we can show

C + € that ^0(0 := log( ) minimizes f{9,c). Let

g{e) := f{do,e) = -{e + c)log(^) + e. c

Then

5(0+) = 0

and

g'ie) = l-log(^)-(e + c)^i

c ^ c = -log( —)<0, Ve>0.

c

Thus 5(€) < 0,V€ > 0. Define .

gni€) ^loge-^°(^)^^£(€^°(^)(^'»-'^P-)),

then gn{€) g(€). Let 9 = 9o{€) in (9), then

00

777 5^ P{ —{Tn,m - npm) > f)

n=l "

00 ^ gglogP(f(Tn,m-np„,)>0

n=l

00

n=l

Given e > 0, 3^£ > 0 such that g{€) + 6^ < 0 and

3iVe e N such that g^ie) < g{€) + 6^,^n > N^.

Therefore

n=l

00 E + E e^3n(c)

n=l n=iV«

00 E + E

n=l

< 00 {by assumption).

00 m

Hence > P(-(Tn — npm) > e) < 00, Vt > 0. n=l

By a similar reasoning we can show that

00 P{-{Tn,m - npm) < -() < OO, > 0. n=l

26
3 Limits of bootstrap distributions : when normalizing constants are unknown

Suppose that assumptions on Xi,X2, - • ,Xn and

Yi,Y2t • • lYjn given in section 2 hold and suppose that an and are unknown. Let

and bm are estimators of and bm based on

Xi, X2, •

•

' , Xn- Now, define the bootstrap distri- bution of a~^{Xn:n — bji) with estimated normalizing constants by

Hn,m{x,Uj) := P{a;;,\Ym:m " bm) < x|X„).

The same choice of m(n) as in the case of known

normalizing constants gives the same results on the

consistency of Hn^jn{x,u) if dm and bm are correctly

chosen as shown in the next theorem. Let = [^]

and k'^ = [-^]. L em J Theorem 4 Assume that Fe D{A). Define

am = F-\l - - F-\l - h

em m

bm = F-\l - -) = Xn-kr.:n. m

If m=o(n), then

sup \Hn,m{x,Uj)-Aix)\^0, (10)

in probability. Moreover, ifYl'^=z\ <

VO < A < 1, then (10) is true w.p.l .

Theorem 5 Assume that Fe -D(^a). Define

m

bm = 0.

If m=o(n), then

sup \Hn,m{x,Uj)-^Ux)\^0, (H)

in probability. Moreover, < oo, VO < A < 1, then (11) is true w.p.l .

Theorem 6 Assume that Fe -D(*a)- Define

dm = ^Fn — (1 ) = Xn:n ~ -^n-fc„:n? m

bm — ^Fn — ^n:nIf m=o(n), then

sup |^„,^(ar,a;)-$«(a:)HO, (12)

in probability. Moreover, ifY^^=\^^

VO < A < 1, then (12) is true w.p.l .

Note that the result of each of the above theorems

implies that

sup I Hn,m{x,L0) - Gn{x) |-> 0,

xeR

in probability if m=:o(n) and w.p.l if Xl^i -^^ <

oo, VO < A < 1. Therefore Hn,m{x,u;) approximates Gnix) uniformly when n ^ oo. Note also

that ni=o(j^|^) is sufficient for Yl'^=i ^"^ <

VO < A < 1.

The following theorem shows that the joint dis- tribution of a~^{Xn:n - bn), a~'^{Xn-l:n " &n),

,a~^{Xn-r+i:n — ^n) cau be bootstrapped successfully.

Theorem 7 Assume the hypothesis on F and

choose dm and bm os in Theorem 4i5)Or 6 according to the domain of attraction F belongs to. If m=o(n), then

sup I P{d:;;^(Ym:m-bm) < Xi, d:;;^^{Ym-l:m-'bm) < X2,

' ' '

> {Ym—T+l-.m ~ bm) ^ ^^rlX^}

-Fr{Xi,---,Xr) HO, (13)

in probability, where supremum is taken among ev- ery xi > • • • > Xr. Moreover, if Yl^=i <

VO < A < 1, then (13) is true w.p.l .

Theorem 7 and the continuous mapping theorem

gives the following.

Theorem 8 Assume the hypothesis on F and

choose dm and bm as in Theorem 4j5,or 6 according to the domain of attraction F belongs to. Let

27
/ : R'' R' 6e continuous a.e. with respect to Fr{',' • • ,•). If m=o(n), then

sup I P{f{a^ {Ym:m-bm),a^ (Ym-l-.m - bm) ,

' '

,

y€R'

{Ym-r+l:m " ^m)) < y|X„}

a-\Xn-r+l:n " 6^)) < Y} H 0, (14)

m probability. Moreover, ifYln=\ < oo, VO < A < 1, then (I4) is true w.p.l .

Corollary 1 Assume that F£ D{'^a)- IJm=o(n),

then

Y — Y

sup I ^ < A^n)

-^( y^Z'y' <x)HO, (15)

^n-.n -^n— l:n m probability. Moreover, ifYlnLi <

VO < A < 1, then (15) is true w.p.l .

Confidence intervals for Op based on the asymptotic distribution of {Xn-.n - ^F)l{Xn:n - ^n-l:n) were considered by Ref. [12]. We apply above

corollary to approximate the critical points of

P{{Xn:n " dF)/{Xn:n " ^n-l:n) < x). Let

Rn —

Ymim Xji'Yi

Y - Ym— l:m First we obtain a large number, say N of bootstrap

replicates of size m(n) from Fn (the empirical dis- tribution of Xi, X2, • • •

, Xn) and then compute R^ i for e = 1, 2, • • • , iV and set

1=1 As 00, H^^^{x) approximate P{Rn < x\'Kn)

which is close to Pi{Xn:n- ^F)/{Xn:n- Xn-l-.n) <

x) when n ^ 00. Thus a 100(1 — 77)% approximate

confidence interval for Op will be

(-^Ti:n '^n,l—Ti2{XTi:n ~ -^n— l:n)? •

X-n-.n ~ ''n,??! (-^n:n -''^n— l:n))5

where r„^^j and r„^i_^2 ^^e chosen such that

where

^1 > 0, 7/2 > 0, 771 -I- r/2 = 77.

4 Conclusions

The proofs of the results in section 3 are in Fukuchi

(1994). We are currently working on extending the

present work to the case of stationary sequence of

random variables under appropriate mixing conditions. Also some simulation work is in progress to

assess the role of the resample size m(n). We are

grateful to Professor S. Lahiri and A. Vidyashankar

for several useful discussion and comments.

References

[1] Efron, B., Bootstrap methods-another look at the jackknife, Ann. Statist., 7 (1979), 1-26.

[2] Athreya, K. B., Bootstrap of the mean in the

infinite variance case, Ann. Statist., 15 (1987),

724-731.

[3] Arcones, M. and Gine, E., The bootstrap

of the mean with arbitrary bootstrap sample

size, Ann. Inst. H. Poincare., 25 (1989), 457-

481.

[4] Singh, K., On the asymptotic efficiency of

Efron's Bootstrap, Ann. Statist., 9 (1981),

1187-1195.

[5] Kiinsch, H. R., The jacknife and the bootstrap for general stationary observations, Ann.

Statist., 17 (1989), 1217-1241.

[6] Bickel, P. J. and Freedman, D. A., Some

asymptotic theory for the bootstrap, Ann.

Statist., 9 (1981), 1196-1217.

[7] Angus, J., Asymptotic theory for bootstrapping the extremes, Commun. Statist.-Theory

Meth.,, 22(1) (1993), 15-30.

[8] Deheuvels, P., Mason, D. and Shorack, G.,

Some results on the influence of extremes on

the bootstrap, Ann. Inst. H. Poincare., 29

(1993), 83-103.

[9] Haan, L. de. On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Math. Centre Tracts, 32 (1970),

Mathematical Centre, Amsterdam.

28
[10] Resnick, S. I., Exterme Values, Regular Variation, and Point Processes. Springer, Berlin,

(1987).

[11] Leadbetter, M. R., Lindgren, G. and Rootzen,

H., Extremes and Related Properties of

Random Sequences and Processes. Springer,

BerHn, (1983).

[12] Cooke, P. J., Statistical inference for bounds

of random variables, Biometrika., 66 (1979),

367-374.

[13] Fukuchi, J., The Bootstrap Method for Extremes of Random Variables. Ph.D. dissertation under preparation, Iowa State University,

(1994).

I

I

29

Extreme Analysis Of Wave Pressure And Corrosion For Structural Life Prediction

Ayyub, B.M.

University of Maryland, College Park, MD

Extreme analysis can be used in structural life expectancy assessment. In this paper, extreme

analysis was used for this purpose in two aspects of life expectancy assessment. These aspects

are (1) extreme wave pressure prediction, and (2) extreme corrosion estimation. Then, they were

used in a time variant reliability assessment formulation of a marine structure. The result is the

reliability of a structure as a function of time, which can be viewed as the cumulative distribution

function of structural life. The presented methodology was performed in an effort to assess the

life expectancy of patrol boats. In the applications discussed in this paper, the underlying parent

distribution in the extreme value analysis was assumed to have an infinite exponential tail. This

assumption can significantly affect the resulting extreme value distributions and assessed

structural reliability levels. In dealing with waves or corrosion, the tails are limited based on the

physics of both problems. The former is limited by the hydrodynamics of waves, and the latter is limited by the size of a corroded element. The effects of limiting the tails of parent distributions on the results of these applications require further investigation.

1. INTRODUCTION

The factors that affect the life of a structure include

design parameters, design safety factors, design

methods, type of strucmre, structural details,

materials, construction methods and quality, loads,

maintenance practices, inspection methods, and other

environmental factors. These factors have different

types of uncertainty that can be classified as: (1)

physical randomness in magnitude and time of

occurrence, (2) statistical uncertainties due to using

limited amount of information in estimating the

characteristics of the population, (3) model

uncertainties due to approximations in the prediction

models, and (4) vagueness in the definition of the

factors, system and/or assessing their/its effect on

life. Therefore, the estimation of life expectancy is a complex process. Because of the stochastic nature of

many of the uncertainties, a probabilistic approach,

as opposed to a deterministic approach, is better suited for life expectancy prediction. Life

expectancy associated with failure modes such as

yielding, plastic deformation and buckling can be

estimated using the extreme analysis, and life expectancy associated with failure modes such as

fracture and fatigue are estimated using the

cumulative value modeling approach. In dealing

with corrosion, life expectancy assessment can be

based on extreme analysis. Example applications of

life expectancy prediction are provided by Ayyub et

al [4], Ayyub and White [3], Harris et al [5], and

Yazdani and Albrecht [12].

2. STRUCTURAL RELIABILITY

ASSESSMENT

The performance function that expresses the

relationship between the strength and load effects of

a structural member according to a specified failure

mode is given by

M= g(Xi,X2,...,Xp)= R-L (1)

31
in which the Xj, i = l,..,p are the p basic random

variables which define the loads, material properties

and other structural parameters, g(.) is the functional

relationship between the basic random variables and

failure (or survival); R is resistance or strength; and

L is load effect. The probability of failure can be

evaluated by the following integral:

Pf= JJ • l^xi^h ^2^ Xp)dxidx2 ... dxp(2)

where fx is the joint density function of Xj, X2,

Xp, and the integration is performed over the region

where M < 0. The strength (or resistance) R of a

structural component and the load effect L are generally functions of time. Therefore, the

probability of failure is also a function of time. The

time effect can be incorporated in the reliability

assessment by considering the time dependence of one or both of the strength and load effects.

3. EXTREME ANALYSIS OF WAVE

PRESSURE

Extreme values based on observational data are very

important in structural safety and life assessment. The prediction of future conditions, especially

extreme conditions, are necessary in engineering

planning and design. The prediction is performed

based on an extrapolation from previously observed

data. For a set of observations (x^, X2, . ., xj^) from

an identically distributed and independent set of

random variables (Xj, X2, Xj^), the distribution of

Xj is called the parent (or initial) distribution. It has

the cumulative probability distribution function

Fx(x) and the density probability function fx(x)-

The maximum value of the observed values is a random variable Mj^ which can be represented as

Mi^ = Maximum (Xj, X2, Xj^) (3)

The exact cumulative and density probability

distribution functions of the maximum value are, respectively, given by

ik

FMj^(m) = [Fx(m)] (4a)

fM,(m) = k[Fx(m)]^-^fx(m) (4b)

It can be shown that for relatively large values of k,

the extreme distribution approaches an asymptotic

form that is not dependent on the exact form of the

parent distribution; but, it depends on the tail

characteristics of the parent distribution in the

direction of the extreme. The central portion of the

parent distribution has little influence on the

asymptotic form of the extreme distribution. For

parent probability distributions of exponential tails, the extreme distribution approaches an extreme value

distribution of double exponential form as k->oo. For

example, a normal or lognormal probability

distribution approaches a Type I extreme value

distribution as k-^o=. In this case, the difference

between an exact distribution for Mj^ and the Type I extreme value distribution is relatively small. The

difference diminishes as k—>oo. Practically, the

difference is negligible for k larger than

approximately 25.

For the purpose of life prediction, the mathematical

model for the extreme distribution needs to be a

function of k in order to relate the outcome of the

analysis of extreme statistics to time. Extreme value

distributions, like the Type I largest extreme value

distribution, are used in this paper to model extreme

load effects. Since the mathematical model is not

sensitive to the type of the parent distribution, as long as it is within the same general class, the

mathematical model used in this chapter is based on

an parent distribution that follows the class of normal

probability distributions.

For a normal parent probability distribution of the

random variable X with a mean value |Li and standard

deviation a, the cumulative distribution and density

functions of the largest value Mj^ of k identically

distributed and independent random variables (Xj,

X2, X-0 are, respectively, given by

Ff^^(m) = Expj-Exp

fMj^(m) =

l(m-ii-aui^) (5)

r«k] Y

Exp

V 0 J A

Exp< -Exp

where

and

ak= [21n(k)]0-5 (7a)

uk = ttk - { ln[ln(k)] + ln(47r) }/(2ak) (7b)

32
The mean value and standard deviation of Mj^ can be

determined approximately using the central and

dispersion characteristics of Type I extreme value

distribution. They are given, respectively, by the

following: — ya

Mean value, Mk = cru]^ + )! + (8)

K <J Standard Deviation, a-\A = —f= (9)

k V6 ay.

The constants n and y have the values of 3.141593

and 0.577216, respectively.

3.1 Example

The method is illustrated by considering the plastic

deformation failure mode of a marine vessel (Ayyub

and White [3], and Ayyub et al [4]) from which this example is taken. Although this example deals with

random sea loads on marine vessels, the method is equally applicable to other random loads, such as wind loads and earthquake loads. For illustration purposes the critical failure mode is assumed to be

plastic plate deformation of the shell of the vessel. The objective of the analysis is to determine the

cumulative distribution function of structural life for

this failure mode.

The end of structural life of the vessel according to the specified failure mode is defined as having to replace more than five plate panels in a specified

critical region at the end of any inspection period. It

is assumed that plate panels are to be replaced when

the ratio of plastic deformation to plate thickness is greater than or equal to 2.0. This assumption is usually based on the resources allocated for repair

and steel replacement for the vessels in their lifetime

maintenance cycle. The inspection schedule of the

boat includes the warranty inspection at the end of

the first year followed by regular inspections every I years, where I can be either one or two years.

In this case, the performance function takes the

following general form:

g = Resistance - Still Water Load - Dynamic Load(lO)

Each of the terms in the above equation are expressed in units of pressure. The still water load is the hydrostatic pressure at the depth of the critical region. It can be determined based on the design

draft. The dynamic load is the extreme dynamic

pressure based on the results from full scale

experiments conducted on one of the vessels. The

resistance term is an empirical expression developed

by Hughes [6] based on elasto-plastic plate response.

In this example, only loads and load effects in headseas are considered. No other heading is considered

because reported stress records by Purcell et al [7]

indicate that they result in much smaller stresses than

the head-seas condition. Eight combinations of

vessel's speed and sea-state for the head-seas

condition are considered herein. These combinations

are summarized in Table 1. For the eight

combinations, strain measurements at locations of

interest were performed by Purcell et al [7]. The

combination of high sea-state and high speeds was

not tested. The percentages that are shown in Table

1 for each combination represent the percentage

usage of the vessel in the corresponding speed/seastate combination. These percentages are based on a survey conducted by the same researchers. The total of the percentages in the table is about 20%, which is the expected percent usage in head-seas.

The performance function as given by Eq. 10

includes two components of pressure, i.e., still-water

and dynamic pressure. The stress due to the still- water pressure component can be modeled using

random variables. Since the stresses due to still- water pressure were not measured, the mean value of

the still-water pressure was determined based on

hydrostatic analysis using the vessel's draft and was

found to be 2.667 psi (Purcell et al [7]). The

coefficient of variation and distribution type of still- water pressure are assumed to be 0.20 and normal,

respectively.

I

33

I
Table 1 . Combinations of ship speed and sea state

Sea State Ship Speed

[Wave

Height]

Low (12

knots)

Medium

(24 knots)

High (29

knots)

Low

[3 ft]

Case 1 (12 knots,

3 ft) 4.0%

Case 2

(24 knots,

3 ft) 1.7%

Case 3 (29 knots, 3

ft) 1.0%

Medium Case 4

(12 knots,

8 ft) 4.7%

Case 5

(24 knots,

8 ft) 1.3%

Case 6

(29 knots, 8

ft) 0.7%

High

[10 ft]

Case 7

(12 knots,

10 ft) 5.3%

Case 8

(24 knots,

10 ft) 1.0%

Not

considered

The strains due to the dynamic pressure were

measured, and the computed stresses should be

modeled using the statistics of extremes. The parent

distribution for the measured stress was assumed to be the probability distribution of a random variable

that is defined as the maximum stress due to dynamic

pressure in 30-second interval for all the cases in Table 1, except Case 8. For Case 8, the interval is taken as 10 seconds. The statistical characteristics of

the parent distribution of stress for the eight cases were determined using the data reported by Purcell

et al [7]. The mean values and coefficients of

variation (COV) for Cases 6 and 8 were based on 10

and 23 maximum values taken from 10 and 23

records of stress time-history, respectively. Other

cases were based on one record each. Then, plate

theory and finite element analysis were used to determine the mean value of the maximum dynamic

pressure, Mean(Pi^^), that causes the measured

stresses. The results are summarized in Table 2. It is reasonable to assume that the COV of the maximum

dynamic pressure, COV(Pj^ax)' the same as the

COV of the maximum measured stress. The mean

value and COV of the extreme pressure were, then,

determined using Eqs. 8 and 9 for an example

vessel's usage period of 15 years at a rate of 3000

hours per year and according to the percent use

presented in Table 1 . The results are shown in Table

2. It was also assumed that the extreme pressure

follows Type I largest extreme value probability

distribution.

Table 2. Statistical characteristics of pressure (15

years of usage)

Case no. Mean

'^^max)

(psi)

COV

'^max^

No. or

intervals in life,K

Mean

^^extrm^

(psi)

COV

^^extrm)

1 1.75 0.0993 216000 2.55 0.0177

2 1.89 0.0993 91800 2.71 0.0186

3 1.99 0.0993 54000 2.83 0.0192

4 6.17 0.0993 253800 8.99 0.0175

5 6.76 0.0993 70200 9.66 0.0189

6 3.07 0.0993 37800 4.35 0.0196

7 7.63 0.0993 286200 11.13 0.0174

8 13.37 1.0121 162000 74.30 0.0477

It is evident from the Table 2 that Case 8 is the most

critical sea state/boat speed combination. Therefore,

for this case the statistics of the maximum and

extreme pressures were determined using the usage

periods of 0.2, 0.5, 1, 2, 5, 10, 15, 50, and 100 years

using Eqs. 8 and 9 as shown in Table 3.

Table 3. Statistical characteristics of pressure for case 8

Usage

period

(years)

Number of

intervals in

life, K

Mean

(^extrm)

(psi)

COV

(^extrm)

0.2 2160 60.49 0.0732

0.5 5400 63.70 0.0657

1 10800 66.02 0.0610

2 21600 68.24 0.0569

5 54000 71.07 0.0523

10 108000 73.13 0.0493

15 162000 74.30 0.0477

50 5400000 77.67 0.0435

100 1080000 79.54 0.0414

The statistical characteristics of the strength of the

material used for the vessel and the dimensions of a

34
plate within the critical region were determined by

Ayyub and White [3]. The mean values and COV of

the yield stress and modulus of elasticity of the

material were estimated to be 47.8 ksi, 29,774 ksi,

0.13 and 0.038, respectively. The mean values and

COV of the thickness and the overall dimensions of

the plate were estimated to be 0.161 in., 11.75 in. x

23.5 in., 0.01, 0.05 and 0.05, respectively.

The failure probabilities of a plate according to the

limit state of Eq. 10 can be determined using Monte

Carlo simulation with variance reduction techniques

(Ayyub and Haldar [2]). The average probabilities of

failure of a plate (Pfp), coefficients of variation of

the estimate of the probability of failure COV(Pfp)

and the numbers of simulation cycles for different

usage periods of the boat are shown in Table 4. The critical region for the vessel was defined as the

region that consists of a total of 28 plates. These

plates were assumed to experience the same loading

and have approximately the same strength

characteristics; therefore, have approximately the same probability of failure. Since the end of life is defined as failure of more than 5 plates (out of the 28

plates), the vessel (or strucmral system) can be

considered to fail if 6 plates or more out of the 28

fail. Let us first consider a warranty period of one

year and an inspection interval of one year. For a

period of one year, plate failure probability (Pfp) is 0.06765 (from Table 4). Since end of life is defined

as failure of at least 6 out of 28 plates, we can

consider the n-out-of-N system with n = 6 and N =

28. Failure probability of this system depends on the

statistical correlation between the plate failures. An

upper bound failure probability is obtained when the

correlation coefficient is unity and a lower bound is obtained when the correlation coefficient is zero. The correlation between the plate failures is assumed

to be small, and so the lower bound is closer to the

acmal (unknown) value. The lower bound failure

probability can be based on the binomial distribution.

The probability of failure of at least six plates out of

28 plates (Pf6/28,l) of one year of service

as 0.00989. Similar calculations for inspection

intervals of two years (I = 2) with Pfp = 0.09403

(from Table 4) yields Pf6/28,I = 0.042719. Since the

warranty period is one year (W = 1), Pf6/28 W

~

0.009895.

Table 4. Probability of failure of a plate (without

inspection effect)

Usage Number of Probability COV

period simulation of failure (Pfp)

(years) cycles Pf

r

0.2 3000 0.03004 0.0490

0.5 3000 0.05092 0.0401

1 3000 0.06765 0.0351

2 3000 0.09403 0.0294

5 2000 0.13950 0.0284

10 2000 0.17200 0.0244

15 2000 0.20310 0.0215

50 2000 0.27760 0.0155

100 2000 0.32900 0.0121

Failure probabilities at different durations of service

T (years of usage) were computed at T = 1, 3, 5, 11,

21, 31, and 51 years were computed and plotted in Fig. 1. This graph of failure probability versus time

is also the cumulative distribution function of

structural life. It is evident from Fig. 1 that by

reducing the inspection interval, expected structural

life can be enlarged. This due to the fact that at the

end of each inspection interval, any reported

deformation damage is to be fixed before sending the

vessel for the next usage period.

0.8

,^

'—

s

0.6 O

o 0.4

tu

abil CD 0.2 o

0

20 40

Structural life (years)

60

1-year inspection 2-year inspection

Figure 1 . Probability of failure based on plate

deformation

35
4. EXTREME ANALYSIS OF CORROSION

One of the problems facing an engineer when

attempting to perform life prediction of a structure is how to adequately deal with corrosion. There are handbooks available which are full of test data for a wide variety of metals, exposure duration's, and types

of corrosive attacks (Schumacher [8]). The difficulty comes in attempting to fit one of the examples to the

real case at hand. Typically, the information on

available corrosion rates is not the type of

information needed by the engineer. The engineer

needs to make a determination of remaining strength

of a panel of plating based on mean thickness and a determination of the integrity of the plating from the

depths of pits. White and Ayyub [9] developed an approach for both planning the number and location

of measurements to be taken using semivariogram

analysis (Ayyub and McCuen [1]), and for using the

information obtained to do a reliability-based service

life assessment of the structure (White and Ayyub

[10]). In this paper, a means of determining a maximum value of pitting depth based on thickness

measurements is incorporated by treating the growth

of pits as a random process with some specific

statistical characteristics.

When performing a service life analysis of marine

structures both pitting and general wastage need to be included in assessing a number of potential failure

modes. The determination of the rate of corrosion

and the rate of pitting has been a major difficulty in designing cost-effective and reliable structures. The

extreme depth of pits can be estimated by (1) the

theory of extremes, or (2) sampling from largest pits.

These two methods are described in the following

sections. The methods serve different objectives.

The first method can be used in cases were the

corroded side of the metal is not accessible.

Therefore, general sampling can be used to determine

the statistical characteristics of thickness. The

resulting probability distribution of thickness can be

treated as an underlying parent distribution in the

theory of extremes. In cases that involve accessible

corroded sides, both methods can be used. However,

the second method provides a direct measurement of

pitting depth. Then the concept of percentile and

largest depths can be used to characterize the

extreme depths.

4.1 Measurements Taken Without Knowledge of

Extent of Pitting

Consider a set of n observations (xi, X2, Xn) from

identically distributed and independent set of random

variables (Xi, X2, Xn). The distribution of Xj is called the initial (or parent) distribution, that has the

cumulative probability distribution function Fx(x)

and the probability density function fx(x). The

minimum observed value is a random variable Mi

which can be represented as Ml = Minimum (Xi, X2, Xn) (11)

The exact cumulative and density probability

distribution functions of the minimum value are given by, respectively:

Fm, (m)= 1-[1-Fx(m)]" (12)

fM, (m)= n[l-Fx(m)r-4x(m) (13)

It can be shown that for relatively large values of n, the extreme value distribution approaches an

asymptotic form that is not dependent on the exact form of the initial probability distribution; but, it depends on the tail characteristics of the initial distribution in the direction of the extreme. The

Type I extreme value distribution is used in this paper to model extreme corrosion. Since the

mathematical model is not sensitive to the type of the

initial distribution, as long as it is within the same

general class, the mathematical model used in this study is based on an initial distribution that follows

the class of normal probability distributions.

For a log-normal initial probability distribution of the

random variable X with a mean value p. and standard

deviation a, the cumulative distribution and density

functions of the smallest value Mi of n identically

distributed and independent random variables (Xj,

X2, Xj^) are of a smallest-extreme Type I distribution, and are, respectively, given by

Fmi (m)= 1 -Exp{-Exp[(ai/a)(m-|i-aui)]}(14)

fMi (m) = (ai/a) Exp[(ai/a)(m - |i - o ui)]

Exp{-Exp[(ai/a)(m-|i- aui)]} (15)

where

ai = [ 2 ln(n) (16)

ui = -ai + { ln[ln(n)] + ln(47t) }/(2ai) (17)

The mean value and standard deviation of Mj can be

determined approximately using the central and

36
dispersion characteristics of Type I smallest-extreme

value distribution, and are , respectively, given by

Mean value, = a uj + |i - y c/aj (18)

Standard Deviation,

J = (tu/V6 Xa/aj) (19)

In using this method, n can be assumed to represent an approximate number of pits in a location of

interest.

4.2 Measurements Taken in Deepest Pits

In the pervious section, thickness sampling can be

performed in the form of a grid that cover a specified

section of a structure. The resulting statistical characteristics were considered to constitute the

moments for a parent distribution with an

exponential tail (e.g. normal distribution). Then, the

theory of extremes was used to determine the

statistical characteristics of the smallest-extreme

thickness as a measure of the deepest pit. In this section, the depth of k pits in a specified location of a

structure are sampled. The statistical characteristics

of these pits can then be determined using the sample

of size k. Assume that the section of interest of the

structure has n pits, which is sufficiently large, and

also assume that the depth of a pit is a random

variable X with the following probability density and

distribution functions:

fx(x) = X exp(-^x) where x > 0 (20)

Fx(x) = 1 - exp(-?ix) (21)

The parameter X can be determined based on the

sampled mean value X as ?l = 1/X. Then, the

deepest pit P in the section of interest has,

respectively, the following cumulative distribution

function Fp(p) and density function fp(p):

Fp(p) = exp[-n exp{-X p)] (22)

fp(p) = n ^ exp(-A, p) exp[-n exp(-A, p)] (23)

Integration or simulation can be used to determine

the mean value and standard deviation of the deepest

pit.

4.3 Example

This example was taken from a study performed by

White and Ayyub [11]. The data used in this example were the results of an ultrasonic hull inspection of one of the vessels of the Class being

smdied (White and Ayyub [9]). There were over

3,000 individual thickness measurements taken on

the roughly 82-foot long hull. The measurements

were reported on a shell-expansion drawing with

large sections marked to indicate areas of excessive

corrosion or pitting. Figure 2 provides an excerpt

from that drawing showing a section consisting of the

plating between transverse frames covering three

longitudinals. In this part of the vessel the frame

spacing is 60-inches and the longitudinal spacing is 24-inches. For this 2880-sq.in. area ten thickness

measurements were taken. Table 5 provides the

locations of the measurement points with respect to the lower left comer of Figure 2. As can be seen from Figure 2, all of the measurements were larger

than the nominal design plating thickness of the hull. This fact was noted for all parts of the hull and not

just this section. The apparent discrepancy in the

measured values when compared to the design

thickness alone would lead one to suspect the

measurements and probably discard them as not being useful. How then can the information provided

by the ultrasonic survey be effectively used?

48.00 n

•20" 244 .25*5

Design Nominal

Thicloiess

• .1875

.240

. .236 240 -248

2^8

' 64.00 in

Figure 2. Measured Thicknesses on Example Bottom

Plating Panel

37
0.1 0.12 0.14 0.16 Thickness

Figure 3. CDF and PDF of Extreme Smallest

Thickness

Table 5. Thickness Measurements and Locations for

Example Plating

Point

No.

x-Location

(in)

y-Location

(in)

Thickness

(in)

1,2 4, 32 4,5 0.231,0.228

3,4 60,6 3,22 0.244, 0.236

5,6 33,61 23, 23 0.240, 0.248

7,8 2,4 30, 43 0.240, 0.204

9, 10 31,60 42, 42 0.244, 0.255

The apparent thickness increase over the design

thickness may be the result of the measuring device

(or operator) not being able to either get good contact with the surface, or misinterpreting the display of the

returning signal, or even a mis-calibration of the

measuring device. In each of these cases we can

treat the difference in the thickness as a bias error. By taking the largest thickness measured and

subtracting the nominal design thickness we can get an estimate of the bias. Later we could vary the size

of the bias to see what effect it has on the final

results. For the example case, an estimated bias of

0.07 inches was used. The ultrasonic measuring

device, if properly calibrated and used by an

experienced operator is capable of providing a very

high accuracy. The accuracy, however, is rapidly

degraded depending on surface conditions, what is behind the plating being measured, and the skill of

the operator.

In performing the extreme analysis, the values for n

to be used in Eqs. 14 to 19 and the confidence level

desired for the confidence interval computations are needed. Because the area under investigation was

identified as being "heavily pitted", a large number

of pits was assumed to be present. Though value of n

has little effect on the results once n exceeds about

100, we will use n = 1000 just to indicate severe

pitting. Figure 3 shows the PDF and CDF (from Eqs.

14 and 15) for the extreme smallest value based on

the parameters for this example.

A confidence interval can now be computed. For this example, a 90% confidence interval was desired as well as a 90th percentile extreme smallest thickness

(largest pit depth). For pit depth the value is used in the inverse CDF of the extreme to fmd that extreme

smallest thickness with only a 10% probability of

being exceeded (having a pit deeper). The results of

the analysis are presented in Table 6.

Table 6. Results of Analysis for Example Plating

Estimated Mean Thickness 0. 1670 in.

Confidence Interval: Upper 0.1733 in. Bound

Lower Bound 0.1607 in.

Depth of Extreme Pit 0.1332 in.

5. CONCLUDING REMARKS

In the applications discussed in this paper, the

underlying parent distribution in the extreme value

analysis was assumed to have an infinite exponential

tail. This assumption can significantly affect the

resulting extreme value distributions and assessed

structural reliability levels. In dealing with waves or

corrosion, the tails are limited based on the physics

of both problems. The former is limited by the

hydrodynamics of waves, and the latter is limited by

the size of a corroded element. The effects of

limiting the tails of parent distributions on the results of these applications require further investigation.

REFERENCES :

[1] Ayyub, B.M. and McCuen, R.C. Optimum

Sampling for Structural Strength Evaluation, J. of

Structural Engineering, ASCE,116(2) (1990) 518-

535.

38
[2] Ayyub, B.M. and Haldar, A. Practical structural

reliability techniques, ASCE, J. Struct. Engr., 1 10(8)

(1984), 1707-1724.

[3] Ayyub, B.M., and White, G.J. Life expectancy

assessment and durability of the Island-Class patrol

boat hull structure, Report Submitted to U.S. Coast

Guard R&D Center, CT (1987).

[4] Ayyub, B.M., White, G.J., and Purcell, E.S. Estimation of structural service life of ships, Naval

Engineers J., 101(3) (1989), 156-166.

[5] Harris, D.O., Lim, E.Y., and Dedhia, D.D.

Probability of Pipe Fracture in the Primary Coolant

Loop of a PWR Plant, Volume 5, Probabilistic Fracture Mechanics Analysis, NUREG/CR-2189

(1981).

[6] Hughes, O.F. Design of laterally loaded plating - uniform pressure loads, SNAME J. Ship Res., 25(2)

(1981), 77-89.

[7] Purcell, E.S., Allen, S.J. and Walker, R.J. Structural analysis of the U.S. Coast Guard IslandClass patrol boat, SNAME Trans., 96(7) (1988), 1- 23.

[8] Schumacher, M. Seav^ater Corrosion Handbook,

Noyes Data Corp., Park Ridge, NJ, (1979).

[9] White, G.J. and Ayyub, B.M. Semivariogram and

Kriging Analysis in Developing Sampling Strategies,

In: Proceedings, Symposium on Uncertainty

Modeling and Analysis, MD, Ed. B. Ayyub, (1990),

360-365.

[10] White, G.J. and Ayyub, B.M. A Probabihstic

Approach for Determining the Effect of Corrosion on

the Life Expectancy of Marine Structures, In: Proceedings, Marine Strucmral Inspection,

Maintenance, and Monitoring Symposium,

SSC/SNAME, VA, (1991).

[11] White, G.J. and Ayyub, B.M. A ProbabilisticBased Methodology for Including Corrosion in the

Structural Life Assessment of Marine Structures,

Report EW-04-92, Engineering, U.S. Naval

Academy, MD, (1992).

[12] Yazdani, N., and Albrecht, P. Risk analysis of

fatigue failure of highway bridges. J. of Structural

Engineering, ASCE, 113(3) (1987), 483-500.

39

Record Values From Rayleigh And Weibull

Distributions And Associated Inference

Balakrishnan, N.

McMaster University, Hamilton, Ontario, Canada

Chan, P.S.

The Chinese University of Hong Kong, Shatin, Hong Kong

In this paper, we study the upper record values from a Rayleigh distribution and derive explicit expressions for the means, variances and covaxiances. We also establish some recurrence

relationships for the single and product moments. These results are then used to derive explicitly the best lineai unbiased estimators for the scale-parameter as well as the location-scale

parameter cases. Some associated inference with regard to the prediction of a future record

value and the test for spuriosity of the current record values are also developed. Next, we

present two examples and illustrate all these inference procedures. Finally, we extend all these

developments to the Weibull distribution and present the necessary explicit algebraic formulae.

1 Introduction

Record values and associated statistics are of importance in many real-life situations involving data relat- ing to weather, sports, economics, and hfe-tests. The

statistical study of record values started with Chandler [11] and since then have been pursued in different directions by several authors; for example, see Click

[15], Galambos [14], Resnick [20], Nagaraja [18], Nevzorov [19], Ahsanullah [2], Arnold and Balakrishnan

[4], and Arnold, Balakrishnan and Nagaraja [5]. The

record values from the exponential distribution and

the best linear unbiased estimators of the location and

scale parameters based on them have been discussed

by Ahsanullah [1]. The prediction of future record

veilues has been discussed for the exponential case by

Dunsmore [13]. Some work of this nature has been

carried out for the extreme value distribution by Nagaraja [16, 17] and Ahsanullah [3], and for the normal

distribution by Balakrishnan and Chan [8].

In this paper we consider the upper record values

from a Rayleigh population and derive explicit ex- pressions for the means, variances and covariances.

We also establish some simple recurrence relationships

for the single and product moments. These results are similar to those established recently by Balakrishnan and Ahsanullah [6] and Bedakrishnan, Ahsanullah

and Chan [7] for the exponential and Gumbel distri- butions, respectively. The latter problem has been

treated exhaustively in the order statistics context by

Barnett and Lewis [10]. Next, we derive explicitly

the BLUEs for the parameters in the one- and two- parameter models. The BLUEs are then used to de- velop prediction intervals for the future record val- ues and also a test for spuriosity of the record value

just observed. Next, we consider the data set given

by Dunsmore [13] and also a simulated data set and

illustrate all the necessary formula/C in explicit algebraic forms. Finally, we extend all these results to the

Weibull distribution and present the necessary exphcit

algebraic formulae.

2 Record values and properties

Let Xt/(2)i • • • be the upper record values aris- ing from a sequence {X,} of i.i.d. Rayleigh variables

with pdf

f{x)=xe-''''\ x>0 (2.1)

and distribution function = 1 - e-^'/2^ x>0. (2.2)

41
Then it is known that the pdf of the nth upper record

value is given by

1

X > 0, n= 1,2,... (2.3)

and that the joint density function of Xu(m) and

Xu(n) is given by

r(n) Jo \2j

(with u =

Tin)

1 -log

1-F{x)

r(m)r(n - m)

1 - F{y)

1 - F(x)

m-l

+ log 1 - F(x)

n — m — 1

0<x<t/<oo, m=l,2,..., m < n.(2.4)

Let us denote £'(X^(„p by a^^^ Var(X(/(n)) by /?„,„, ^W(m)> ^C;(n)) by ' Cov(Xc;(^), Xu(^n)) by

/?m,n- For convenience, we will also use a„ for an ^ and

ctm.n for am',ri We then have the following theorems.

Theorem 1 For n = 1,2, ... , and k = 1,2,

^(fc) _ gfc/2 r("+ 2)

" " r(n)

end /or 1 < m < n_,

r(m+i)r(n + l) r(m) r(n +

i)'

consequently,

(2.5)

(2.6)

(2.7)

(2.8)

and

Cov{Xmm),Xu(^n)) = 2

T{n+l)

r(m+i)

r(m)

r(n+|)

r(n+i) r(n)

Proof: From (2.3), (2.1) and (2.2), we have

4'^ = ^°°x'=|-log[l-F(x)]| f{x)dx

(2.9)

Next, from (2.4) we have for 1 < m < n m-l

m,n r(m)r(7

2 2 \ fi— iTi—

1

X I y - y j ye-y^l^dxdy

1 2'"-ir(m)r(n - m)

2 \ n — m — 1 (2.10)

where

2 \ n — m—

1

(2.11)

du

By setting u = x^/y^, (2.11) becomes

l(y) - /^V"'(i-t/)"-'"-^-^

Jo ^v" = ^y''"+'5(m+i n-m), (2.12)

where B{a,b) = T{a)T{b)/T{a + 6) is the complete

beta function. Substituting the expression of /(y) in (2.12) into (2.10), we have

B{rn + |, n — m)

2'"r(m)r(

n r

n — m)jQ

g-yV2^2m+2

n—m— 1 ydy

2'"r(m)r(n- m) Jo

(setting t; = yV2)

2B{m + i,n - m)

= 2

r(m)r(n — m)

r(m + ^)r(n + 1) r(m)r(n+|)

r(n + 1)

Formulae in (2.7)-(2.9) then readily follow.

For the Rayleigh distribution, it is easily observed that

x/(x) = 2(- log [1 - F(x)l I

[1 - F{x)] . (2.13)

42
By using this relation, we establish below some simple

recurrence relations satisfied by the single and product moments of record values. Similar results for the

exponential and Gumbel populations are due to Balakrishnan and Ahsanullcih [6] and Balakrishnan, Ahsanullah and Chan [7], respectively.

Theorem 2 For n = 1,2,..., and k = l,2,...,

.ai'\ (2.14)

Proof: Let us consider

(t) _ k + 2 n

Jo

[1 - F{x)]dx (using (2.13)).

Upon integrating by parts, we obtain

2 kT{n) i-f(i) f{x)dx

log 1 - F{x)

n-l

f{x)dx

2r(n+ 1)

2n

T

kT{n)

"n+l "n (2.15)

Then, (2.14) follows by rearranging (2.15).

Theorem 3 For k,l = 1,2.. ., and m > 1 2m

and for 1 < m < n — 2

(2.17)

Proof: For proving (2.16), let us consider from (2.4)

m-l

1 - F{x)

f{y)dxdy/T{m)

n

I -Fix)-

1 f°° / y'f{y)liy)dy, (2.18)

"i) -/o

1 - F{x)

m— 1

where

(using (2.13)).

Upon integrating by parts, (2.19) yields

2

fix)

1-F{x)

1-F{x)

dx

dx

(2.19)

Ky) = y*|-log[l-F(y)]|

£r'=|-log[l-F(x)]|

F{x)

dx

Substituting the above expression of I{y) into (2.18), we obtain

m,m+l ^°°/+'|-log[l-F(y)]} f{y)dy

^m —

1

/CO /-y -m / / ar*=y'^-log 1 - F{x)

Jo Jo

I - F(x)

^(^^^"'^^

2m a+i) (k,i) k

r(m)

(2.20)

Relation in (2.16) is simply obtained by rearranging

(2.20).

Next, for proving (2.17), let us write for 1 < m < n —

2

(2.16) a(^'^^ =

] ry'ny)Iiy)dy, (2.21)

i {m)L {n — m) Jq

where

J(y) = ^'x'^j-log

|-log|^l-F(y)

J-log[l-F(y)

m-l

1 - F{x)

+ log

1 - F(x)

n — m—

1

1 - F{x) dx

1 - F{x)

+ 1 - F{x)

n — m— 1 dx.

43
Upon integrating by parts, 7(y) can be written as

(n — m — 1) / Jo1 - Fix)

fix)

1-F{x) -log l-F(y)

+ log|^l-F(x)

-m^ x*|-log

n — m —

2

dx

1 - Fix)

+ log

-log

I -Fix)

1 - Fiy)

m-l

N n — m—

1

1 - Fix) I dx (2.22)

Substituting the expression of /(y) in (2.22) into Eq.

(2.21), we obtain

"m+l,n "m,n (2.23)

Relation in (2.17) is simply obtained by rearranging

(2.23).

If we interchange the order of the double integration

in the proof of Theorem 3 and proceed along the same

lines, we can derive the following relations.

Theorem 4 For k,l = 1,2, . .

. , and m= 1,2,...

(M) _[±l^(fc,0

"m,m -a^'+'A + 2 — 2 "i.m + l "'l"m+l,m+2 "m+ 1 )'

(2.24)

and for 1 < m < n — 2 _ l + 2in-m) (kj^ m,n + l 2(n - m)

m

n — m

(2.25)

3 Inference for the one-parameter Rayleigh distribution

Suppose the first n record values

from a one-parameter Raylei]gh distribution with pdf

fiy; (T) = ^exp(--^y x>0, OO, (3.1)

are available. Then, by following the generaUzed least- squares approach, we may derive the BLUE of a

(Balakrishnaoi and Cohen, [9], pp. 74) as

«=i and

/here

a' Ha

Y - iyu{i),yu{2)>---,Yu{n)y,

a = (ai,Q;2, . . . ,a„)'

(3.3)

and

« = ((A,;)).~/=l,2 n = ((^ij)).J = 1.2..,.,n.

Since j is in the form p,g_,- for i < j where

and

fi can be written explicitly as

F 1 (P2''l ~ Pi 93)

' = J = 1,

(Ptii-i-p.-i^iXpi+iii-Pifli+i)

^o-l i = = n <ln(rn<ln-l-Fn-11n)' ' ' _1_ Pi+l'7i-Pi9i+l

j = i+ l,i=l,2 n-1,

l> - i| > 2- (3.4) That is,

9

2' + 1 = — —, J = 2,3, ... ,n - 1,

OJn,n = (2n-l)

9n

wi.i+i = -(2i-H), i= l,2,...,n- 1, = <^i,i+i, i = 1,2,- ,n- I,

i^ij = 0, lj-i|>2.

Eqs. (3.2) and (3.3), when simplified, simply yield

Y,

(3.5)

44
and

Var(o-') = (7^ nr^(n)

r2(n+i) 1 (3.6)

Prediction of the future record: Suppose the up- per records ^(/(i), yc;(2), • • •

,

yu{m)^^ = 1.2,... have been observed. Then the

best linear unbiased predicted value of the record

yu(n), n > m + 1, can be written as

Yuin) = ^'«n (3.7)

where a' is the BLUE of cr based on m records. In- stead of the predicted value, one might be interested in the prediction interval for ^[/(n) with a certain confi- dence. The prediction interval for ^[/(n) be based on the scale-invariant statistic

a"

(3.8)

where once again is the BLUE of a based on the

first m records. Using (3.5), we can rewrite the statistic rf„ in (3.8) as

Tin = am {yU(Ti) - yU(m)} /yU{m) _ ^ jXniiii 1 - ""-[yu^r'

Since y^/2cr^ is distributed as a standard exponential

variable, we can easily show that ^{/(m) /ydin) is dis- tributed as a beta (m, n — m) variate (for example, see Dunsmore, [13]). Thence, the 100% prediction inter- val for yu{n) is obtained to be

yU(_m),yU(^m)

where Bq is the lower a percentage point of the beta

(m, n — m) distribution.

Test of spuriosity of the current record: Suppose ^[7(1)1 • • ) ^[/(n-i) have been observed and a new

record ^[/(n) has just been observed. Sometimes we may be interested in testing for the spuriosity of the

current record value yu{n)- For this purpose, we may

use the scale invariant statistic yu{n)

)/V^]

rpo

•^In —

or* (3.9)

where tr' is the BLUE of a based on the first n — 1 records; using (3.5), therefore, T°„ in (3.9) becomes

Tr„=a._i-^^. (3.10)

At a level of significance, we will conclude the current record yt/(n) to be a spurious record if yu(n) is greater

than yu(n-i)/V^> where is the lower a percentage

point of the beta (n — 1, 1) distribution, namely, Bq =

4 Inference for the two-parameter Rayleigh distribution

Let us now suppose that the first n upper record values

^1/(1)) ^1/(2))

• •

. yu{n) from a two-parameter Rayleigh distribution

with pdf

f{y,l^,<^) = e , /i < y < oo, (T > 0,

(4.1) are available. Once again, by following the generalized

least-squares approach, we may derive the BLUEs of fx and <T (David, [12], pp. 130; Balakrishnan and Cohen,

[9], pp. 80-81) as

n n

;i* = ^

a,yt;(i) and (t* = ^biYu(^i), (4.2) 1=1

?here

and

with

i=l

a' fta.1' ft -a' flla' ft ^ ~ {a' na){l' ni) - (a' ni)2 ^ ^

b =

1' ma' n-r naV a

{a' aa){V ftl) - (a' ^1)2'

l' = (l l---l)ix„

(4.4)

and a and fi as defined in the last section. The

variances and covariance of the above estimators are (David, [12], pp. 130; Balakrishnan and Cohen, [9], pp. 81) given by

Var(/i°)

(72 Var((r')

a' na

and

Cov(fjL* ,a*)

(a' na){l' ftl) - (a' fil)2

r ni

(a' Uoc){V ni)-(a' ftl)2

a' ni

,(4.5)

,(4.6)

(a' na){l' ni)-{a' 01)2

(4.7)

Omitting the intervening algebra, we get the coeffi- cients of the BLUEs of fi and cr from (4.3) and (4.4) to be

a; =

2 On^nA - 1 OCnqn

(2i)(a„g„A - 1)

, i = 2,3,...,n-l

= 1 2(a„9nA - 1)

n-1

3 +

1=2

45
and

bi

bn

where

A

9n

2a„g„A- 1'

1 qn 2za„9„A - 1

, z = 2,3,...,n- 1,

j=2

gn-1

L 9n - 1

The variances and covariance of these estimators are obtained from (4.5)-(4.7) to be

and

Var(M*)

(72 Var((T*)

Cov(/i*,<T*;

an9nA - 1

ttn^nA - 1

Qn

0-2 Q!n9nA - 1 Prediction of the future record: Suppose the upper records Yu{i),yu{2),

^t/(n-i))(" > 3) have been observed. Then the best

linear unbiased predicted value of the record ^(/(n) can be written as

(4.8)

where fi* and cr* are the BLUEs of fi and a based on

the first n — 1 record values. Suppose we are interested in giving an interval for

Y(/(n) with a certain confidence. This prediction inter- val for ^(/(n) n^a^y be based on the location and scale invariant statistic

^2n —

cr* (4.9)

where o" is once again the BLUE of a based on the

first n — 1 upper record values. For aiding the users, we have determined some percentage points of the

statistic in (4.9) through Monte Carlo simulations

(based on 10,001 runs). These simulated percentage

points of are presented in Table 1 for n = 3(1)11.

With the help of this table, one could easily construct 100(1 — Qr)% prediction intervab for the future record

value Yxj{n)-

Testing for spuriosity of the current record:

Suppose the upper records Y(7(i), yc;(2), • • • , ^c/(n-i)

have been observed, and a new record Yu{n) has just

been observed. We may sometimes be interested in testing for the spuriosity of the current record value

Yu(n). For this purpose, we may use the location and

scale invariant statistic

a*

(4.10)

where [i' and a' are once again the BLUEs of \i and

<y based on the first n — 1 records. Large values of

will support the spuriosity oiYu(^y

For assisting the users, we have simulated some crit- ical points of the statistic in (4.10) through Monte

Carlo simulations (based on 10,001 runs). These simulated percentage points of T^^ are presented in Table 2

for n = 3(1)11. By using this table, one could easily

test for the spuriosity of the current record value Yv{n)

at the desired level of significance.

5 Illustrative Examples

Example 1: Dunsmore [13] has given the size of rock

crushed by a rock crushing machine. The machine has

to be reset if, at any operation, the size of rock being

crushed is larger than that has been crushed before. The following are the records of the sizes dealt with

up to the third time that the machine has been reset:

9.3,24.4,33.8.

Suppose for illustration that the sizes of the rocks to be crushed can be represented by independent one- parameter Rayleigh random variables. A simple plot

of these three upper record values against the expected

values of the Rayleigh upper record values indicates a

strong correlation (correlation coefficient as high as 0.984). Hence, the assumption that these record val- ues have come from the Rayleigh model seems quite

reasonable. From formula (3.2), the BLUE of <t is simply obtained to be

a —

33.8 ^3

14.38.

The 90% prediction interval for the fourth upper

record is obtained to be

33.8, 33.8/\/0. 11/3 33.8,49.6

Furthermore, if the fourth upper record has been observed and it is not in [33.8,49.6], we can conclude

46
that the observed record is a spurious record at 10%

level of significance.

Now, suppose the upper records are assumed to have come from the two-parameter Rayleigh model.

We then compute the BLUEs of and <t to be

^l' = (2.000 X 9.3) + (0.333 x 24.4) - (1.333 x 33.8) = -18.33

and

<t' = -(0.8511 X 9.3) - (0.1418 x 24.2)

+(0.9929 X 33.8) = 22.21. From Table 1, we determine the 90% one-sided pre- diction interval for the fourth upper record to be

33.8,69.1 .

If the fourth record has been observed and it is greater

than fj.' +c' X 3.93 = 68.96, we can conclude that the

record is a spurious record at 10% level of significance.

It is of interest to mention here that the above given

prediction interval [33.8,69.1] is very close to the pre- diction interval based on the exponential distribution

given by Dunsmore [13]. This is not surprising as the

Rayleigh distribution is also seen to fit the data very

well.

Example 2: For the purpose of illustration, we simulated a set of record values from the Rayleigh distri- bution with /i = 50 and <t = 10. The following are the

simulated upper record values:

66.42, 72.27, 78.07, 81.82, 86.33, 87.42, 90.05. The BLUEs of y, and a are computed in this case to be

H' = 1.2245 X 66.24 -j- 0.2041 x 72.27

-1-0.1361 X 78.07 + 0.1020 x 81.82

+0.0817 X 86.33 + 0.0680 x 87.42

-0.8163 X 90.05 = 54.54

and

(t' = -0.3332 X 66.24 - 0.0555 x 72.27

-0.0370 X 78.07 - 0.0277 x 81.82

-0.0222 X 86.33 - 0.0185 x 87.42

+0.4942 X 90.05 = 9.67.

The 90% one-sided prediction interval for the next upper record (eighth record) is obtained as

[90.05,90.05 + 9.67 x 0.733] = [90.05,97.14].

6 Results for the Weibull Distribution

All results for the Rayleigh distributions developed in the previous sections csui be extended to the Weibull

distribution with pdf

f{x) = x'-^e-^'/', x>0, OO (6.1)

and cdf

F(x) = 1 -e-^'/^ 2:>0,c>0, (6.2)

where c is the known shape parameter. Analogous to Theorem 1, we then have the following theorem for the Weibull distribution.

Theorem 5 For n = 1,2,..., and k = 1,2,..

r(n + i)

Tin)

and for 1 < m < n

^^.2/o r(m+i)r(n+f)

r(m) r(n + M'

(6.3)

(6.4)

consequently,

E{Xu(n)) = c i/c/£(!Lti)

I r(n)

Var(Xc/(n)) = c2/cf r(n+f) ^ r(n+i)

Tin) Tin)

and

Cov(X(y(^),Xc/(„))

^r(m +

i) rr(n+f) r(n +

i)

Tim) lr(n+i) Tin)

(6.7)

For the Weibull distribution, the following relation is easily observed:

x/(i) = c|-log I -Fix)

I

I -Fix) . (6.8)

By using (6.8) and proceeding along the same lines as in Theorems 2-4, we can establish the following recurrence relations for the Weibull distribution.

47
Theorem 6 For n = 1,2, and k = 1,2,

(jfe) k + nc /t.-) ^ nc

(6.9)

Theorem 7 For k,l = 1,2..., and m= 1,2,.

^-.-+1- Ar + mc'^^+i (6.10)

and for 1 < m < n — 2

Theorem 8 For k,l = 1,2, . .

.

, and m > 1

m,m+2

l+C

m,m+l — ma

an(/ for 1 < m < n — 2

(6.12)

/ + c(n-m) .^ ^^ (k,!) _ . ^ ^

n — m,„ \ "m+ l,n + l "m+ l,n

(6.13)

From Theorem 5, we observe that the variancecovariance matrix ((Aj)) of the upper record values can be written as Piqj,i < j, where

- .i/c£(i±i)

Pi = c

and

9; = C^/^ r(i + f) r(i + i)

,r(i + i) r(;)

Therefore, using (3.4), the inverse of ((Aj)) can be

explicitly written as (C+1)2

r(i + !)'

r(z)

[c2(2z"2 - 2z + 1) + c(4f - 2) + 1]

i = 2,3,...,n- 1,

,-2/c r(n)

r(n+f) gn X [(nc - c -f 1) (nc - c + 2)]

,

= _c-2/<=,^^zc(ic+l),

f = l,2,...,n- 1, W.J = 0, |j-i|>2.

One-parameter Weibull model: Suppose the first n upper record values Yu(i),Yu(^2),- ,Yu(n) from a

one-parameter Weibull distribution with pdf

fiy;<^) = ^exp(-£^y y>0, <T>0, (6.14)

are available. The BLUE of a can be derived, using

(3.2), as

''^^"-^ a = (6.15)

with its variance as

Var(a*) = a' r(n)r(n+f)

r2(n+i) - 1 (6.16)

Two-parameter Weibull model: Suppose the first n upper record values >c/(i),yi7(2), • • • ,yt/(n) from a two-parameter Weibull distribution with pdf

/z < y < oo,cr > 0, (6.17)

are available. The BLUEs of fj. and a can be derived,

using (4.3) and (4.4), as

and

where

ai =

a; —

a„ = 1

anqn c-^/"(c+l)

anQn-l r(l+i) '

an9nC-^/'(c-l) r(0

Qn^nA-l r(l+f)

z = 2, 3, • •• , n — 1, ,-2/c

(6.18)

(6.19)

a„<?„A - 1

_(^±I)_ + (e-l)V^Xi)_

and

c-2/=g„(c + l) (a„g„A-l)r(l + f)' _ c-^/-g„(c-i) r(0

^' - - a„,„A-l r(z + f)'

^-2.3,---,n-l,

48
bn =

,-2/c 9n C+ 1 a„g„A- 1 \r(l + f)

n-l

with

A = c-2/'<!

-^^^

+ fc - 11 T —

n-l

nil

+

r(n)

r(n + f)

(nc - c + l)(nc - c + 2) 9n -

1

The variances and covariance of these estimators are obtained from (4.5)-(4.7) to be

and

Var(/z-)

(72 Var(cr')

Q:n9nAn - 1

'

a^qnA - 1

'

9n

0-2 Q:„9nA - 1

Inference procedures for the WeibuU distribution can

be developed along the Unes of Section 3 and 4 by

maJcing use of the exphcit forms of the BLUEs pre- sented above. For brevity, we have not pursued this here. Acknowledgments

The first author would like to thank the Natural Sci- ences and Engineering Research Council of Canada for funding this reseaxch.

References

[1] Ahsanullah, M. (1980). Linear prediction of

record values for the two parameter exponential

distribution, Ann. Inst. Statist. Math. 32, 363-

368.

[2] Ahsanullah, M. (1988). Introduction to Record

Values, Ginn Press, Needham Heights, Massachusetts.

[3] Ahsanullah, M. (1990). Estimation of the parameters of the Gumbel distribution based on the m

record values, Corn-put. Statist. Quart. 3, 231-

239.

[4] Arnold, B. C. and N. Balakrishnan (1989). Relations, Bounds and Approximations for Order Statistics, Lecture Notes in Statistics 53,

Springer-Verlag, New York.

[5] Arnold, B. C, N. Balakrishnan, and H. N. Nagaraja (1992). A First Course in Order Statistics,

John Wiley & Sons, New York.

[6] Balakrishnan, N. and M. Ahsanullah (1993). Relations for single and product moments of record

values from exponential distribution, J. Appl.

Statist. Sci. (To appear).

[7] Balakrishnan, N., M. Ahsanullah and P. S. Chan

(1992). Relations for single and product moments of record values from Gumbel distribution,

Statist. Probab. Lett. 15, 223-227.

[8] Balakrishnan, N. and P. S. Chan (1993). On the

normal record values and associated inference.

Submitted for publication.

[9] Balakrishnan, N. and A. C. Cohen (1991). Order Statistics and Inference: Estimation Methods, Academic Press, Boston.

[10] Barnett, V. and T. Lewis (1984). Outliers in Statistical Data, Second edition, John Wiley Sons,

New York.

[11] Chandler, K. N. (1952). The distribution and fre- quency of record values, J. Roy. Statist. Soc, Ser. B 14, 220-228.

[12] David, H. A. (1981). Order Statistics, Second edition, John Wiley & Sons, New York.

[13] Dunsmore, L R. (1983). The future occurrence of

records, Ann. Inst. Statist. Math. 35, 267-277.

[14] Galambos, J. (1978). The Asymptotic Theory

of Extreme Order Statistics, John Wiley &

Sons, New York. (1987). Second edition, Krieger,

Florida.

[15] GUck, N. (1978). Breaking records and breaking

boards, Amer. Math. Monthly 85, 2-26.

[16] Nagaraja, H. N. (1982). Record values and ex- treme value distributions, J. Appl. Probab. 19,

233-239.

[17] Nagaraja, H. N. (1984). Asymptotic Unear prediction of extreme order statistics, Ann. Inst.

Statist. Math. 36, 289-299.

49
[18] Nagaraja, H. N. (1988). Record values and re- lated statistics - A review, Commun. Siaitsi. - Theor. Meih. 17(7), 2223-2238.

[19] Nevzorov, V. B. (1987). Records, Teoriya Veroyatnosiet i ee Premeneniya (Theory of Probability and Its Applications) 32(2), 219-251 (in Russian).

[20] Resnick, S. I. (1987). Extreme Values, Regular

Variaiton, and Point Processes, Springer- Verlag,

New York.

50
Table 1. Simulated percentage points of the statistic for the two-parajneter Rayleigh distribution

n 0.01 0.025 0.05 0.10 0.90 0.95 0.975 0.99

3 0.0048 0.0123 0.0238 0.0520 3.7321 7.6296 14.8732 38.4384

4 0.0041 0.0098 0.0199 0.0413 1.5834 2.4319 3.7907 6.5884

5 0.0033 0.0086 0.0177 0.0380 1.1380 1.6498 2.2644 3.3609

6 0.0034 0.0085 0.0166 0.0332 0.9045 1.2637 1.6990 2.3337

7 0.0032 0.0071 0.0137 0.0273 0.7326 0.9934 1.2950 1.7487

Table 2. Simulated percentage points of the test statistic T°

for the two-parameter Rayleigh distribution

n 0.900 0.950 0.975 0.990 0.995

3 5.6121 9.5095 16.7532 40.3184 75.8375

4 3.9335 4.7821 6.1409 8.9401 12.6876

5 3.8797 4.3915 5.0061 6.1027 7.0952

6 3.9888 4.3480 4.7833 5.4181 6.1317

7 4.1750 4.4947 4.8057 5.2405 5.7279

51

On The Estimation Of The Pareto Tail-Index

Using /c-Record Values

Berred, A.M.

Universite du Havre, Le Havre Cedex, France

Let {Xn, n > 1} be an i.i.d. sequence of positive random variables with a continuous

distribution function F having a regularly varying upper tail. In this paper we consider the

k-record versions of two statistics introduced in [1] and study their asymptotic behavior.

Such statistics can be used as alternative estimates for the exponent of regular variation of

the tail 1 — F.

1. Introduction

Let {Xn, n > 1} be an i.i.d. sequence of positive random variables having a continuous distri- bution function F with regularly varying upper

tail. Namely

(F) 1 - F(x) = x-'^l°'L{x), for X > a > 0,

where q > 0 and L is a slowly varying function at

infinity. Denote by Xi,n < • • • < Xn,n the order

statistics associated to the sample Xi , . .

. , Xn . Let

k be a positive integer and define the sequences of

the k-record times and values (Ref. [2], [3]) by

T^''\i) = min{i > r^^)(i- 1),

R{j) >j-k-\-l},

where R{n) is the sequential rank of in the sample Xi, ... ,Xn , i.e., Xn = Xji^n),n, forn > 1 (for

the general theory of records Ref., e.g., chap. 6 of

[4], chap. 4 of [5], [6], [7], [8] and the references

therein). We consider the k-record versions of two

statistics (for other statistics based on record val- ues Ref. [9], [10]) introduced in [1],

Q„ = - {logX(^)(n) - logX('=)(n - m)} ,

fin = -r-,T.^ogX^'^\n-i + l),

n(m)

^

where l</;<7i, l<m<n and n(m) = nm — m{m- l)/2.

The statistics a„ and /3„ can be used as esti- mates of a. For the estimation of a based on ex- treme values, Ref., e.g., [11], [12], [13], [14], [15],

[16], [17], [18], [19], [20]. The paper is organized as follows. Section 2 is devoted to the main results concerning the consistency and the asymptotic normality of aJid

The proofs of these residts are given in section 3. Section 4 contains some examples which illustrate

the results of section 2. Finally, section 5 presents some numerical results concerning the behavior of

the estimates a„ and in practice.

2. Results

In the sequel, we will impose some assumptions on the increase of the sequences {rrin^ ^ > 1} and n > 1}.

(Ml) m = rrin — oo is a sequence of integers

such that 1 < m < n and m/n —> 0

as n —> 00

,

(M2) m = rUn —> oo is a sequence of integers

53
such that 1 < m < n and m/ log n —> oo with m/n —> 0 as n —> oo. (K) k = kn —> oo is a sequence of integers

such that 1 < k < n and k/n —> 0 as n —> 00.

Whenever one of the assumption (Ml), (M2) or (K) is made, we write k and m instead of k^ and

TTin for the sake of simplicity.

The notations " = ", "-i", and

" —^ a.s. " stand respectively for equality in dis- tribution, convergence in distribution, convergence

in probability and almost sure convergence.

Denote by h{x) = — log(l - F{x)) the cumulative hazard function associated to F. The function h is nondecreasing on (— oo,+oo), so that its inverse function may be defined by

H{x) = h^{x) = mi{t : h{t) > x} on (0,+oo).

We now state our main results concerning the

limiting behavior of the above statistics. Theorem 1. Assume that (F) and (K) hold.

Then

(i) under (Ml) with k = 0{m),

a as n —> oo

(ii) under (M2) with k = 0{m),

a. a a.s. as n —> oo.

Theorem 2. Assume that (F) and (K) iioid. Then for I < m < n fixed or for m = rrin sat- isfying (Ml),

/5n a a.s. as n oo.

Furthermore we need the following assumptions on the slow variation of L to ensure the

asymptotic normality of and Pn

(SRI) VA > 1,

(SR2) VA > 1,

(SR3) VA > 1,

LjXt)

m

LjXt)

m

L{Xt)

Lit)

- 1 = Oig(t)) ast —>oo,

- 1 ~ K{X)g{t) ast —>oo,

- 1 = o{g{t)) as t —> 00,

where ^ is a positive function such that g{t) —> 0 as t —> 00. The assumptions of slow variation with

a remainder term (SRl-3) were introduced in [21]

(for a general review on slow variation with a re- mainder and its applications Ref. [22]).

Theorem 3. Assume tliat (F), (Ml) and (K) iio]d

with k = 0(m) and m/y/n —>^ 0. Then if L is (SRl-3) with g nonincreasing and

y/mg

^ "fc"* )) 0 as n —> oo, then

(2.1) :/^{an-a) -^AfiQ,l) asn oo. Theorem 4. Assume that (F) and (K) hold.

If I < m < n and k/y/n —>• 0 (n —>ooj or m, = TUn satisfies (Ml) with k — 0{m)

(n —^ooj, L is (SRl-3) with g nonincreasing and

V^giH •Jn—r 0 as n —» 00, then

(2.2) ^iPn-c^) -^MO,l) as n—^oo.

a

3. Proofs

Let {en, n > 1} be a sequence of i.i.d. unit ex- ponential random variables and denote by Sn =

ei + • • • -\- Cn, n > 1, their partial sums. The fol- lowing lemma gives a representation of the exponential k-record values in terms of the partial sums

Sn, n> 1. Lemma 1. Let k be a positive integer, then

{e('=)(n), n > 1} i {Sn/k, n > 1}.

Proof. The Theorem 1 in [3] (Ref. also Lemma 1 in [2] ) implies that the sequence {e(^)(n), n > 1}

is a Markov chain with transition probabilities

P (e^''\n + I) > x\e^''\n) = yj =

f exp{-k{x -y)}, x > y, \ ^, X <y.

and initial probability

P (e^''\l) > a;) = €xp{-kx), x > 0.

Since the same holds for the sequence {Sn/k, n >

1}, it follows that

{eW(n), n > l} = {Sn/k, n > 1}.

This establishes our lemma.

54
We may assume, without loss of generality,

that the original probability space carries, in addition to the sequence {X^, n> 1}, a sequence

{cnj n > 1} of i.i.d. unit exponential random variables (since h is continuous, we may take =

h(Xn))- Let {kn, n > 1} be a sequence of positive

integers such that 1 < < n, we consider the

double array ^X^^'^i), i > l,j > ij- of k-records.

Now by Lemma 1,

|e(*")(n - r), 0 < r < n} =

[^f^^ 0 < r < n|.

Since h is continuous it follows that H is strictly

increasing and

d

'Sn- H 0 < r < n

So from now on we shall assume without loss of generality that X^''''\n — t) = H{Sn-r/kn), for

71 > 1 and 0 < r < 71. we state two lemmas related to the slowly varying function L in (F).

Lemma 2. For every slowly varying function I log/(z)

log a: 0 as a; —>^ oo.

Proof. Ref., e.g., [23], Proposition 1.3.6, p. 16.

Lemma 3. Assume that (F) holds. Then

(i) log H{x) = ax + log X'(e^), for 0 < x < oo;

(ii) logi'(x) ~ alog L{R-{x))

Proof. Suppose that (F) holds. We prove (i). Set R{x) = x^/L{x). Note that h{x) = \ogR{x).

It follows that R is regularly varying with a positive exponent — . Hence (Ref., e.g., [23], Theoa

rem 1.5.12 p. 28) its generelized inverse R'~{x) =

x°'L'{x), where L' is a slowly varying function.

Now ^(x) = E^(e^) and

log5'(a:) = Qx + logi'(e''), for 0 < x < oo. We show (ii). Noting that R{R'^{x)) ~ x

as X —> oo, we have L''^(x)/L{R'~{x)) —>• 1 as X —> 00 and

iogi:'(x)

1,

as X

QlogX(E-(x))

OO. Our assertions axe established.

To simplify the proofs, we introduce some notations:

^1 Sn—m

At = Q , m

Al = A|iogi'(exPy)-logi:'(exp%^)},

Bl =

a n{m) fr[

Bl = ^Elogi'(exp^).

^ ' 1=1

It foUows by (i) of Lemma 3 that

(3.1) an=Al + Al,

(3.2) (in = B\^ Bl

we implicitly make use of (3.1) and (3.2) in the

proofs of Theorems 1-4.

Proof of Theorem 1. Assume that (F) and

(K) hold with k = 0{Tn). We first prove that

Al 0 (71 —»• oo) under (Ml) and A^ —» 0 a.s. {n —* oo) under (M2). The Karamata Representation Theorem for slowly varying functions (Ref.,

e.g., [23], Theorem 1.3.1, p. 12) impHes that

(3.3)

Sn—T

r)\ exp

At =

+

A; m Jexp

exp ^

t dt,

where is a bounded function on [a, oo) such that

77(x) —* c (x —>-oo, |c| < oo), and € is a contin- uous function on [a,oo) such that €(x) —>0 as X —>-oo. Under (Ml), the law of large numbers

yields

(3.4)

^

m

(3.5)

^-

m

oo a.s. as n —» oo.

oo a.s. as n —>-oo.

as n —*-oo. Therefore, it PHence Cn —*• 0 a.s remains to show that ^„ 0 under (Ml) and —* 0 a.s. under (M2). write as

(3.6)|en| = -

m // €{e')dt

J

^ti—

m

< m

m

sup

55
Now (3.4) imply that sup s^,^ s„ |e(e*)|

k - - k a.s. as n —>^oo. Since

0,

(3.7) = —, for 1 < m < n,

the assumption (Ml) and the law of large numbers

imply again that

(3.8) 1 as n —> 00. m

Denote by {9n = Cn — 1, n > 1} the centred se- quence of {Cn, n > 1}. The moment generating

function of Oi is

$(i) = ^(exp(i0i),

exp{-t)

1-t , foi t < 1.

Hence the assumption (M2) and the Theorem 2.1 in [24] give

(3.9) 1 a.s. as n •00. m

Finally, relations (3.8) and (3.9) yield respectively

(i) and (ii). The theorem is established.

Proof of Theorem 2. Assume that (F) and

(K) hold. We first suppose that 1 < m < n is fixed. Note that n{m) ~ nm (n —>-c>o), the strong

law of large numbers yields for 1 < i < m,

(3.10)

+1

n(m)

Consequently

Bl1— a.s. as n m

a a.s. as n

00.

oo.

Therefore it remains to prove that B\ —»• 0 a.s. as n —> 00. Since 1 < m < n is fixed, this reduces to showing that

A:logi:^(exp%^)

n{m)

0 a.s. as n —> oo,

for 1 < i < m. Rewriting the argument of the last limit as

(3.11)

logX^(exp^r^)5n-.-+i

logexp^^ifi^ n{m) '

the first term converges to 0 almost surely by

Lemma 2, the second one converges to — almost

m

surely by (3.10).

Assume (Ml). Since

m n ~ m nim) ' j=i ^

'

is a Cezaro mean with the summand tending a.s. to 1 by (3.10), consequently B\ —> q a.s. as n —>• 00. Now write

2 1 A log i:'(exp ^^) mSn-i+i

\ logexp%i±i n{rn)

We see again that B^ is a Cezaro mean with the

summand converging to 0 by (3.11), it follows that

B^ —> 0 a.s. as n —> oo .

Proof of Theorem 3. Assume that (Ml)

holds. By (3.7) and the central limit theorem for

i.i.d. random variables — ({A\-a) -^A/'(0,1) asn—^00.

Now assume that L is (SRI). It is weU known

that (Ref. [22], Theorem 2.2.2) log L has representation

logL{x) = rj(x) +

J^'

0{git))

dt

1

as X —»• 00

,

where Tj{t) = c + 0{g{t) (|c| < oo,x —>-oo). In view of (ii) of Lemma 3), log L' can be represented

as

logL'{x) = v'{x) + 0{g{t))j as x —>oo,

where r)'{x) = c' -\- OigiR-ix))) (|c'| <

oo,a: —>'Oo). Hence

log XV) = c' + 0{g{nix))) +

rm^) dt

= c' + 0(^(F(x)))-f

r OigiHmdlogHit),

A(i)

56
as X —* 00. It follows by (3.3) that

Sn—r

o g H

m

J^Oig{HmdlogH{t),

as n —> 00. Therefore our assertion wiU be proved

if we show that y/m C„ —>• 0 and y/m —>• 0 as n —>oo. Since k = 0(m), g is nonincreasing and

P

0,

we have y/mCn —* 0 as n —> oo. Now we take care

of ^/rn S,n We have

lenl < 0(l)Qn sup g{H{t)),

fc - - fc

Under the assumption that g is nonincreasing, we

obtain

as n —>• 00. Now the assumption

y/mgiH Sn0, as n —>• oo,

and Theorem 1 imply that y/m (n —» 0 as n —^oo. The proof in the cases (SR2-3) is similar to that under (SRI), We therefore omit it.

To prove the asymptotic normality of /?n we

approximate the main term in (3.2) by —

.

n Lemma 4. Assume that (F) holds. Then

(i) ifl < m < n is fixed

Q

n

1 p 1 — I as n —>• 00

;

^ n.

(ii) if (Ml) holds

Bl = -Sn + Op — I as n —^ oo. n \n

Proof. This amounts to evaluating for 1 < i <

m < n,

Sn Sn—i+l

mn n(m)

Sn - Sn-m (m - 1) ^ mn 2n n(m) = Oo — as n

n

•00,

by Markov's inequality. Hence

Bl--Sn <

mn

t=i

Sn Sn—m

n{m)

m{m — 1)

2n n(m)

+

1 = Opy—j if m is fixed ,

as n —>-oo. This completes the proof of the

lemma.

Proof of Theorem 4. Applying Lemma 4 and

the central limit theorem for i.i.d. random variables, we obtain for 1 < m < n fixed or m = m„

satisfying (Ml) with y/m/n —> 0 as n —>-oo, ^(bI-o) -^MiO,l) asn—>oo. We wiU be done if we show that

(3.12) v^B^^Oasn —>oo.

Assume that L is (SRI), 1 < m < n is fixed

and k/y/n —>• 0 (n —*-oo). Since n{m) ~ nm

{n —> oo), it is sufficient to prove that

4= log i^'(exp %i±i) ^

0 as n

y/n k

00,

for 1 < t < m. In view (3.12) and the fact that g

is nonincreasing, we have

log Z,'(exp

/"'"^ k

y/n

c' + O

y/n \^n-»-H

Sn-

) =

+

k

\/n Jh{i) I

" Oiginit)))dlogHit)<

Jh(l)

Sn—m gin +

57
log H(Sn-m/k)

0{V^g{t))dt,

as n oo

as n —> oo. Now the assumption

implies that

(3.13) -^logL'(exp%^)—

0

as n —> 00 and for 1 < i < n. When m =

satisfies (Ml), y/nB^ is a Cezaro mean with the

summand tending to 0 in probability by (3.13),

consequently (3.12) is true. The proof in the cases (SR2-3) is similar.

Example 3. Let 1 - F{x) = cx-i/'* exp[(logx)

for X large and c>0, 0<^< 1. Thus L{x) =

cexp [(logx)^ We see that logZ-(x) ~ (logx)" as X —>oo. Hence if 0 < ^ < 1/2, (2.2) is valid for

1 < < n fixed or sequences k = kn —> oo satis- fying (Kl) with k = o{y/n) as n — oo. Clearly L is (SR2) with g{x) = l/ilogx^-^

and K{X) = OlogX. Hence g(^H(^^

an )

1-9

a.s. as n 00 by (i) of Lemma 3,

Lemma 2 and the law of large numbers. It fol- lows that (2.1) is true for sequences k = kn —>^oo

(

1-20 \

n2-2e

J (n —>-oo,0 <

4. Examples

We give some examples of distributions satisfying

(F) and demonstrate the applicability of the previous results.

Example 1. Let 1 - F{x) = cx-i/"(loga;)^ for x large and c > 0, ^ 7^ 0. It follows that

L{x) = c(loga:)^ and logL{x) ~ ^ log log x as x —»-oo. Therefore (2.2) is valid for 1 < A: < n

fixed or sequences k = kn —^oo satisfying (Kl)

with k = o{y/n) as n —*• 00.

Now L is (SR2) with g(x) = l/\ogx and

K{X) = ^logA. Hence by (i) of Lemma 3,

Lemma 2 and the law of large numbers,

g

^ ^

~ —. Consequently (2.1) is true

for sequences k = kn —>-oo satisfying (Kl) with

k = o{y/n) as n — 00. Example 2. Let 1 - F{x) = x-'^l°'{c + dx'^) as X —>-oo; c, d, ^ > 0. In this case L{x) = c + dx~^

and \ogL{x) —> logc. Hence (2.2) is true for 1 <

k < n fixed or sequences k = kn —> 00 satisfying

(Kl) with k = o^s/n) as n —» 00. Here L is (SRI) with g{x) = x~^ but it is more convenient to consider directly y/k A^. It

follows that y/k = ^p(~^) —*°°)'

the law of large numbers. Therefore (2.1) is true

for sequences k = kn —*oo satisfying (Kl) with

k = o{y/n) di.s n —>oo.

5. Simulation and numerical results

To validate the theoritical results of section 2, we

give some numerical results to show how the esti- mates Qn and Pn behave in practice. The k-record

values are obtained by generating exponential ran- dom variables and then applying the function H to

their sums. We consider here the Paxeto distributions of the form l-F(x) = cx^/*^ (l -|- x~^), where

a,c > 0.

5.1. Number of k- records

The expected number of k-record values and its variance are related to the sequence k = kn in the

following manner. Set Tn^^"\n) = knY17=kn^l^

and v(*")(n) = m^^^\n)-kl TJi=kr, 1/^'^ ^ > 1- Denote by N^^'^\n) the number of k-record val- ues in the sequence X\,. . . Then (Ref., e.g., Theorem 3.2 in [6] and Lemma 2.1 in [8]),

E{N^^^\n)) = m^^^\n),

VaT{N^^^\n)) = v^^^\n) for n > 1.

Table 1. The expected number of k-records in a sample of size n for a given sequence k = kn- The

notation [x] stands for the integer part of x.

58
n k — k \ J

10

[log n\ O.oD

[n J 1 9fi

[W J O.O ( U.oo

100

n r\<y 71 1 [log 7ZJ O.Ul

23.58 3.75

37.42 3.61

1000

[log n] 31.21 4.97

108.20 8.81

347.58 12.61

5.2. Consistency of and /3„

The tables 2-4 show that the estimates and Pn

behave quite well for a reasonable number of k- records (see Table 1). For c = 1, the statistic

is more precise than On in estimating a. In the

other hand, when c > 1, the estimate (3^ tends to over-evaluate the value of a for n < 10.

In the tables 2-4 below, n represents the number of k-records, Qn the first estimate of a for a

given n, /3n the second estimate of a for a given n,

<7n the theoritical standard deviation for a given n and (T{Qn or /3n) the standard deviation of 5000

estimates of a.

Table 2. The sequences kn = mn = [logn].

n c a

5 0.469 0.500 0.479

10 1 0.5 0.464 0.353 0.332

15 0.491 0.353 0.349

5 0.973 1.000 0.978

10 1 1.0 0.989 0.707 0.695

15 1.008 0.707 0.719

5 0.988 1.000 1.005

10 2 1.0 0.982 0.707 0.706

15 1.013 0.707 0.731

5 2.047 2.000 2.076

10 1 2.0 1.998 1.414 1.436

15 1.977 1.414 1.376

IV r\kX a On

CO U.OUo n 99*? n 91 1

11 U.OX i U. lOO nU. 1X<J^ ^9

110^ n fin9 u.ouz nu. 1izy9Q

0

1 (\(\A 1.UU4 n A AO

in 1

i. 1 n 1 nn u.oxu u.oxo

n 9^s n 9fi.^ 5 1.135 0.447 0.469

10 2 1.0 1.138 0.316 0.344

15 1.097 0.258 0.276

5 2.009 0.894 0.899

10 1 2.0 1.992 0.632 0.632

15 2.002 0.516 0.520

Table 3. The sequences kn = mn = [nP'^^].

n c a an 0-(0!n)

5 0.471 0.500 0.491

10 1 0.5 0.491 0.500 0.501

15 0.496 0.500 0.501

5 0.481 0.500 0.491

10 3 0.5 0.505 0.500 0.515

15 0.506 0.500 0.510

5 0.964 1.000 0.996

10 1 1.0 0.969 1.000 0.969

15 1.009 1.000 0.984

5 1.979 2.000 1.988

10 1 2.0 2.028 2.000 2.015

15 2.006 2.000 1.983

n c a Pn criPn)

5 0.509 0.223 0.214

10 1 0.5 0.501 0.158 0.163

15 0.499 0.129 0.129

5 0.613 0.223 0.246

10 3 0.5 0.557 0.158 0.170

15 0.538 0.129 0.135

5 1.006 0.447 0.436

10 1 1.0 0.995 0.316 0.314

15 0.999 0.258 0.257

5 2.006 0.894 0.889

10 1 2.0 2.001 0.632 0.627

15 2.004 0.516 0.517

Table 4. The sequences kn = m„ = [ti^-"*^].

59
Leningrad. Otdel. Mat. Inst. Steklov., 153

(1986), 115-121.

[8] Deheuvels, p. and Nevzorov, V. B.,

Limit laws for k-record times, (1992),

Preprint.

n c a <7n 5 0.579 0.353 0.325

10 1 0.5 0.530 0.228 0.0262

15 0.469 0.228 0.276

5 1.068 0.707 0.667

10 1 1.0 1.015 0.577 0.547

15 0.996 0.577 0.583

5 2.014 1.414 1.362

10 1 2.0 1.987 1.154 1.120

15 1.984 1.154 1.140

n c a cr{l3n)

5 0.540 0.223 0.209

10 1 0.5 0.522 0.158 0.155

15 0.508 0.129 0.125

5 1.034 0.447 0.437

10 1 1.0 1.009 0.316 0.313

15 1.002 0.258 0.255

5 2.000 0.894 0.884

10 1 2.0 1.994 0.632 0.635

15 1.997 0.516 0.513

References

[1] Berred, M., On record values and the exponent of a distribution with a regularly varying

upper tail, J. Appl. Probab., 29 (1992a), 575-

586.

[2] DZIUBDZIELA, W. AND KOPOCINSKI, B.,

Limit properties of the k-th record values, Zastos. Mat, 15 (1976), 187-190

[3] Deheuvels, P., The characterization of dis- tributions by order statistics and record val- ues — a unified approach, J. Appl. Probab,,

21 (1984), 326-334.

[4] Galambos, J., The asymptotic Theory of

Extreme Order Statistics, John Wily & Sons,

1978.

[5] Resnick, S. I., Extreme Values, Regular variation and Point Processes, SpringerVerlag, New York, 1987.

[6] Nevzorov, V.B., Records, Theory Probab.

Appl., 32 (1987), 201-228.

[7] Nevzorov, V.B., On the kth record times

and their generalisations. Zap. Nauchn. S'

[9] x'/ET-RED, M., On record values and the es- timation of the Weibidl tail-coefficent, C. R.

Acad. Paris, Serie I, 312 (1991), 943-946.

[10] Berred, M., K-record values and the

extreme-value index, presented at XIIP'

Rencontres Franco-Beiges de Statisticiens,

(1992b), submitted to JSPI.

[11] Hill, B.M., A simple general approach to inference about the tail of a distribution, Ann.

Statist, 3 (1975), 1163-1174.

[12] De Haan, L. and Resnick, S.L, A simple

asymptotic estimate for the index of a stable

distribution, /. Roy. Statist. Soc, 42 (1980),

83-87.

[13] Teugels, J.L., Limit theorems on order

statistics, Ann. Probab., 9 (1981), 868-880.

[14] Hall, P., On some simple estimates of an exponent of regidar variation, J. Roy. Statist.

Soc, 44 (1982), 37-42.

[15] Mason, D.M., Laws of large numbers for sums of extreme values, Ann. Probab., 10

(1982), 754-764.

[16] Haeusler, E. and Teugels, J.L., On

asymptotic normality of Hill estimator for the

exponent of a regular variation, Ann. Statist.,

13 (1985), 743-757.

[17] CsoRGo, S., Deheuvels, P. and Mason,

D.M., Kernel estimate of the tail index of a

distribution, Ann. Statist, 13 (1985), 1050-

1077.

[18] Deheuvels, P. and Mason, D.M., The

asymptotic behavior of sums of exponential

extreme values. Bull. Sc. Math., V Serie, 112

(1988), 211-233.

60
[19] Deheuvels, p., Haeusler, E. and Mason, D.M., Almost sure convergence of the

Hill estimator, Math. Proc. Camb. Phi. Soc,

104 (1988), 371-381.

[20] Deheuvels, P., Haeusler, E. and Mason, D.M., On the almost sure behavior of sums of extreme values from a distribution

in the domain of attraction of a Gumbel law,

Bull. Sc. Math., 2^ Serie, 114 (1990), 61-95.

[21] Smith, R.L., Estimating tails of probability

distributions, Ann. Statist., 15 (1987), 1174-

1207.

[22] GoLDiE, C. M. AND Smith, R. L., Slow variation with remainder : Theory and applications, Quart. J. Math., Oxford (2), 38 (1987),

45-71.

[23] Bingham, N.H., Goldie C.M. and

Teugels J.L., Regular Variation, Cambrige

University Press, 1987.

[24] Hanson, D.L. and Russo, R.P., On the

law of large numbers, Ann. Probab., 9 (1981),

513-519.

61

The Point-Process Approach To The Directional

Analysis Of Extreme Wind Speeds

Bortot, P.

Universita di Padova, Padova, Italia

In this paper the problem of directional modeling of extreme wind speeds is discussed. An adapted

version of the method developed by Smith in 1989 [1], based on a point-process view on extreme

value problems, is proposed. Techniques are considered to solve difficulties deriving from serial correlation and angular dependence. The procedure is illustrated with an appUcation to real data.

1. Introduction

The problem of directional modeling of extreme

wind speeds, although rarely discussed in hterature,

plays an important role in civil engineering. In feet,

the directional analysis of the extremal behaviour of

winds provides engineers with useful information for an accurate choice of building orientation and leads

potentially to considerable savings.

The problem has already been dealt with by Coles

and Walshaw [2] . They employ a modified version of

the r largest annual events model in which the

parameters of the Generalized Extreme Value (GEV)

distribution are expressed as functions of direction. In

this paper our aim is to use recent developments in the

methodolog>- of univariate analysis of extreme values,

suitably adapted, to obtain a model which takes into

account directional aspects of wind process. In doing

this we will partly follow the ideas proposed by Coles

and Walshaw.

2. Description of data

The data analyzed were collected at the

meteorological Military Air Force station in Trieste,

Italy. The station is situated at 8m above sea level; the

anemometer reaches a height of 39m above ground

level. The data consists of measurements of the

direction and the average intensity of wind: averages

are calculated over the ten minutes preceding the

recording which is limited to the so-called synoptic

hours (00, 03, 06, 09, 12, 15, 18, 21). The records cover a period of 23 years: from the 1^ of January

1951 to the 30* of December 1973, nominally 67200

observations. Of these values, 1361 are missing from

periods when the equipment was out of service. Wind

speed is measured in knots. There are 36 directional

sectors: the angle is recorded to tlie nearest 10° in clockwise orientation starting from North.

Like most environmental data, the time series of

wind speed departs from iid sequences in two respects:

first, in being heavily seasonal and second, in exhibiting short-range dependence due to the

persistence of the weather leading to clustering of

high-level exceedances. Seasonahty and serial correlation compel us to adjust the tools derived from

classical extreme value theory, since it concerns maxima of independent, identically distributed random

variables.

To assess the connection between wind intensity and

sector of origin, a boxplot of wind mean speed has

been constructed for each direction (Figures la and

lb). As we are only interested in extreme values, we

have discarded all the observations below 10 knots. It seems clear that the extremal behaviour of wind is sfrongly influenced by direction: the North-East sector being the most affected.

3. The point-process approach

The method employed in this analysis is an adapted

version of the one developed by Smith [1] in the study

63
5of

40

O

10 60 120 180 Degrees from North

(Central bar at median; upper and lower bars al quartiles; whiskers extend from the 10* to the 90* percenule)

Fig. la. Boxplotfor wind speeds in each direction (10°-! 80°)

so

o

20

190 2«0 300 3&0 Degrees from North

(Central bar at median; upper and lower bars at quartiles; whiskers extend from the 10* to the 90* percentile)

Fig. lb. Boxplotfor wind speeds in each direction ( 1 90°-360°)

of ground-level ozone concentrations. It is based on a

point-process approach to extreme value problems,

which was originally introduced by Pickands [3] and,

in more recent years, emphasized in the books of

Leadbetter, Lindgren and Rootzen [4] and Resnick

[5]. In its apphcation to ozone data, the method has

shown a considerable versatility and a wide

apphcability. Moreover, it includes all the other

methods of analysis of extremes (the traditional

method, the Peaks over Threshold method and the

method based on the r largest annual events) as

special cases. Let X^,X^,... denote an iid sequence with common

distribution function F and A/„ = max(A'i,...,A'„).

Suppose that there exist normalizing sequences

a^>0, such that, as n-^cc,

where H is the Generahzed Extreme Value

distribution, i.e.

Hix;M,C7,k) = exp[-{l-k{x-fi)/ay"]

valid over the range {x: I- k(x - > 0};

<7>0, -oo<//<oo and -oo <k< +00. Let ={X^-bJ/a^ and denote the point

process on the plane with points at (;/(« + 1),}V),

/ = 1, n. Then, under a topology which essentially

excludes points whose ordinates approach the lower

endpoint of distribution H, P^ converges, as « -> 00, to a nonhomogeneous Poisson process P with intensity measure

A{(f„r,) X (x,cc)} = (t,-tJ\-k{x-^)/arr (3.1)

whenever 0< r, < < 1 and l-^(x-//)/cr > 0. This result may be regarded as fundamental in yielding all relevant asymptotic distributional

properties. For instance, the limiting conditional

probabihty that Y^>u + y given Y^>u is given by

the ratio between the mean number of points that P

has on (0,1) x (« + ^,oo) and the mean number of

points on (0,1) x (u,oo). By using (3.1) we obtain

[\-k(u + y-/i)/c7f' ky

[\-k{u-n)laf'' ^ a-ku+k/i

which is the Generalized Pareto distribution with

shape parameter k and scale parameter a-ku+k/i.

It should be noted that the parameters of the

intensity measure of the process P are the parameters

of the maximum limiting distribution. In an

apphcation to real data this result enables us to use the

Poisson process P to represent the time series and, in

64
this way, to obtain more precise estimates of the

parameters of the annual maximimi distribution.

Nevertheless, seasonal variation and serial correlation,

inherent in wind data, make it impossible to apply the

approach above described directly, since, so far, it has

been confined to iid sequences.

4. Seasonality and serial correlation

Smith [1] suggests overcoming the seasonal problem

by splitting up the year into a number of periods, each

of which is modelled separately: that is, allowing all the parameters of the process P to be seasonally

dependent. However, to avoid further complicating the

analysis, we choose not to take seasonahty into

account, beheving that this simplification can be

justified.

With reference to serial dependence, since extreme

values tend to occur in clusters, a technique

extensively employed in this kind of application is to

tr\' to identify clusters of high-level exceedances, with

the intention of concentrating on cluster maxima for the rest of the analysis. There is no universally

accepted method for identifying clusters. The one

followed by Smith and adopted here consists of fixing a threshold u and a cluster interval z*. Two adjacent

exceedances of u are deemed to he in the same cluster

if the interval between them is less then z*. If the time

interval between them is longer than z*, it is assimied

that the old cluster has finished and a new one begun.

In this way clusters are defined and only the largest

observation within each of them is retained for fitting.

Cluster interval and threshold are chosen empirically.

The technique that we suggest for this choice is to assess the goodness of fit of the model for different

values of z* and u over a reasonable range and, finally, select the smallest of the couples which yield

satisfactory results. In fact, if a certain value of z* ensures independence between clusters, then this independence is also ensured for all higher values of

z*. Nevertheless, the increment of z* reduces the

number of independent observations available, with

the twofold effect of 1) raising questions about the

vahdity of the asymptotic arguments justifying

approximations based on the Poisson process P;

2) reducing the estimation precision. Similar

arguments are valid for u. The properties of the point-process approach can be

employed to test the goodness of fit of the model on

varying z' and u. A first test is based on equation (3.2)

and consists of graphically assessing how closely the excesses over the threshold « fit a Generalized Pareto

distribution with parameters k and a-ku +k/i. It is possible to carry out an alternative graphical test using GEV distribution to transform annual maxima

to uniformity before plotting them against the

empirical distribution function.

5. Directional components

Following ideas by Coles and Walshaw, we choose

to calculate the directional components of velocity of

each recorded speed and then to model wind

components. This is because each recorded speed,

although associated to a certain direction, has a

contribution from all directions. This procedure is also

justified by the fact that we work with mean speeds,

each of which is the result of speeds coming from a number of different sectors. So, let denote a mean speed of magnitude Y in direction a. The component of velocity of 1^ in direction <p is Ycos(a - ^) if\a-4\ modulo TTKTcjl;

otherwise it is zero. Using the resolution into directional components we

obtain 36 complete series of wind intensities: one for each sector. As with the original time series, the 36

sequences feature a high short range autocorrelation.

6. Direction as a covariate

We could easily apply the point-process approach

to the directional study of Trieste data by fitting the

model described in section 3 separately to the

observations of each of the 36 sequences.

Nevertheless, since data are recorded on a fine directional scale, we can obtain a considerable gain in efficiency modelling the parameters of the process P

(k, // and cr) as fimctions of direction. This also has

the advantage of smoothing annual maximum speed

distributions across directions in accordance with the

features of the physical process. Suppose that the annual maximum wind components

in direction a (aGA = {10°, 20°, 360°}) have

GEV distribution with shape parameter k^, location

parameter //^ and scale parameter a^. As suggested

by Coles and Walshaw, a natural choice is to express

each of the three parameters as a sum of the first terms in a Fourier series. Therefore, let

"1 ." : =«i+Z^'^s(ror-Ci,),

65
"3 <^CC=^3+T.^3, «>S(ra - ^3^, ) . (6. 1)

/=!

The parameters in the model are now a^, b^, and c^,

;

r = 1, n^; m = \, 2 and 3. In order for the model to be well-defined we have to restrict b^>0 and

0°<c^,<360°. We have also to exclude models in which b^=0 for some m and / with the

corresponding c^, 0°, since b^, =0 corresponds to the absence of the t-th harmonic term and, in this case, the value of c„, does not influence the model.

7. Angular dependence

Since we have chosen to model GEV parameters as functions of direction, we can not ignore the

dependence of extreme wind speeds across directions.

This dependence, which Coles and Walshaw call angular dependence to distinguish it fi-om temporal

dependence discussed in section 4, is a consequence of

the feet that storms tend to give successive high

observations in a number of different directions.

Moreover, the resolution into directional components

itself induces dependence across directions.

To solve the problem of temporal dependence we

filter each of the 36 sequences of wind components

following the rule described in section 4 and using a

cluster interval z* = 24 hours and a threshold u = \l

knots. These values derive fi"om a previous analysis

which was carried out adopting the point-process

approach but ignoring directional aspects. They are

the smallest couple ensuring a good fit of the pointprocess model to Trieste data. From this operation we

obtain 36 series of cluster maxima, each being

independent temporally; however, dependence across

directions remains.

We have overcome this further obstacle following

ideas proposed by Smith [6] for dealing with spatially

dependent data. It consists of constructing the

likelihood function as if there were independence

across directions. To account for angular dependence,

standard errors of parameter estimates and likelihood

ratio test are suitably modified.

Let G be the observed information matrix under the

model which assumes independence. If the

independence assumption were vahd, G ' could be

used to approximate the covariance matrix of

maximum likelihood estimates. Smith shows that, to take account of dependence, this approximation should

be replaced by G 'VG"', where V is the covariance

matrix of log-likelihood derivatives. If years are independent, V may be obtained empirically. Similar

arguments are applied to adjust the usual asymptotic

distribution of likelihood ratio test for model

discrimination.

8. Directional model

In adapting the method proposed by Smith to our

study we assume stationarity from year to year, since

the previous analysis, ignoring direction, proved that

there was no (linear) trend in the data. Let M. denote the length of observation in days in year / (z = l, 23), N.^ the number of cluster maxima in year / and direction a (fir 6 A ) and Y^^j

(y = l, N.^) the cluster maxima in year / for direction or. Let w denote the fixed threshold (m = 1

1

knots). Following the discussion of previous sections,

we assume that for direction cx , in any given year, the

exceedance times of threshold u and cluster maxima

form a nonhomogeneous Poisson process with

intensity measure given by (3.1) with Ma'^a K

replacing , a and k. Then, the likehhood function in direction a is:

23 M

L,=Yi[cxp{--^(\-kju- )icj, }

•

/=1 jOj>

• -h^—i^-K (x^ - )K r--')] (8.1)

where k^,fi^ , and are defined by equations (6.1).

Assuming independence across directions, the

log-likelihood is 1 = ^^^\d.L^. Maximum likelihood

estimates of a^, b^, and c^,(r = 1, m=\,2, 3) are obtained by maximizing numerically /. The quantities of greatest interest for engineers are return levels. The 7-year return level in direction a is the l-T"' quantile of the annual maximum

distribution in direction a ; it is given by

It should be noted that qj-^ is the quantile of the

distribution of the annual maximum component in direction a. .

66
0.4

I

02 « 0.0 a

Degrees from North

Fig. 2a. Shape Parameter Estimates

-J 12 -1 :

• , ^ ^ 10 60 110 160 210 260 310 360 Degrees from North

Fig. 2b. Location Parameter Estimates

-\ ' \ \ —r \ r——

r

10 60 110 160 210 260 310 360 Degrees from North

Fig. 2c. Scale Parameter Estimates

9. Results

The procedure followed for identifying the model

which yields the best fit to Trieste data consists of

varying the number of harmonic terms in equations

(6.1) and comparing fitted models using likelihood

ratio tests modified in the manner advocated by Smith.

The model chosen with this procedure has three

harmonic terms for the location parameter, two for the

scale parameter and for the shape parameter. Results

are shown in Table 1. Standard errors of parameter

estimates (shown in parentheses) are adjusted for

angular dependence. The parameters c^, (t = I, m = 1, 2 and 3) are expressed in radians.

It is not possible to reduce the number of harmonic

terms for the three parameters, and, in particular, for the sh^e parameter, without suffering a heavy loss in

terms of goodness of fit ofthe model.

The last question concerns assessing the fit of the

chosen model to data. For each of the 36 sectors separately we have employed the two tests outlined in section 4: one based on the Generalized Pareto

distribution and the other based on the GEV

distribution. The plots (not shown) indicate a good

agreement between expected values and observed

values.

An alternative technique is proposed by Coles and

Walshaw. It consists of examining how closely

maximimi likelihood estimates of k^, ju^ and

follow the corresponding values when each sector is considered separately. Figures 2a, 2b and 2c contain

such comparisons. In each plot lines join the

maximum likelihood estimates and the upper and

lower bounds of the 95% confidence interval, while

the plotted points represent the parameter estimates

when (8.1) is maximized on each sector separately. A

similar comparison has been made for the 50-year

return level (not shown). It is important to recall that maximum likelihood estimates derived firom the

separate sectors analysis have larger standard errors

than those of the covariate model. For this reason one

or more points outside the confidence interval do not

necessarily indicate a lack of fit of the model. The

plots show no systematic deviation of covariate model

estimates from separate sectors analysis estimates.

Therefore, we can conclude that the chosen model

seems able to capture the variations induced by

direction on annual maximum wind component

distribution.

10. Conclusion

In this application to wind data the method proposed

by Smith has confirmed the quahties shown in the study of ozone concentrations. Besides ensuring accurate results and requiring very mild assumptions,

it is a flexible tool and can be easily adapted for handling complex features of r^ data. The analysis has led to the conclusion that direction

heavily influences the extremal behaviour of wind. As

before mentioned, the proposed model seems to accurately describe variations across sectors. Therefore, it can be of great utility to civil engineers

for a corrert assessment of wind impact on structures. As a final comment, we recall that we have only

adjustai the point-process method in order to take into accoimt angular dependence, with no attempt to model

such a dependence. This would require the use of tools derived fi'om multivariate exfreme value theory.

67
Table 1. Maximum likelihood estimates and standard errors ofthe selected model

—

-u.uui (u.uuy)

h , A 1 OA /A A 1 V\

Cu 1 £<0 /A OA1 \ i.oDy (u.zui)

parameter L A AQQ /A A'2/1\ A ;I'2'7 /A 1 0'>\

CI2 T) T)< /A <0«\

02a fi 2'39 /"A ')'3'3\

/^.*

.

1 1 "37 /A A? ')^ ^ 9/;n ICS MAS

parameter l.Tl\ (0.096)

2.287 (0.202)

3.703 (0.084)

3.791 (0.281)

^. 0.711 (0.113)

Scale Cli 6.189 (0.343)

parameter lh2 0.590 (0.137)

CX2 3.047 (0.255)

ACKNOWLEDGEMENTS

I am gratefiil to Prof. Alessandra Salvan and Dr. Matteo Grigoletto, of the University of Padua (Italy), for all their help and support throu^out this project.

REFERENCES

[1] Smith, R.L., Extreme value analysis of

environmental time series: an apphcation to trend

detection in ground-level ozone. Statistical Science, 4

(1989), 367-393.

[2] Coles, S.G. and Walshaw, D., Directional

modelling of extreme wind speeds. Submitted, 1992.

[3] Pickands, J., The two-dimensional Poisson process

and extremal processes, J. Appl. Probab., 8 (1971),

745-756.

[4] Leadbetter, M.R., Lindgren, G. and Rootzen, H.,

Extremes and Related Properties of Random

Sequences and Series, Springer, New York, 1983.

[5] Resnick, S., Extreme Values, Point Processes and

Regular Variation, Springer, New York, 1987.

[6] Smith, R.L., Regional estimation from spatially

dependent data. Submitted, 1991.

68
High Boundary Excursions Of Locally Stationary

Gaussian Processes

Bmker, H.U.

Institut ftir Mathematische und Versicherungslehre, Bern, GermanyLet {X{t), t 6 T} be a locally stationary Gaussian process and /(i), t E T a, continuous

function, where T is a finite or infinite interval. An asymptotic estimate for small

probabilities P(X(t) > f{t), some < G T) is derived by approximating the density of

the first passage time and integrating over T. This work extends a result proven by

J. Cuzick, Ref. [5], for stationary Gaussian processes.

keywords: locally stationary Gaussian process, boundary crossing, first passage time

Let {X{t), t € T} be a Gaussian process (T =

[0,r), r < oo) and f{t), t eT a. continuous function. We are interested in

P(X(i) > f{t), some t £ T). (1)

The asymptotic behavior of this probability has

been studied during the last thirty years.

J. Pickands, Ref. [8], proved the following

Theorem 1 Let {X{t), t > 0} be a separable

stationary Gaussian process with

EX(t) = 0 and covariance function

p{h) = 1 - l/2i22|/i|" + o(|Ar) {h 0)

(0 < R< oo, 0 < a <2) such that p{h)\ogh

0 as h —* oo (Berman's condition).

If u = Ur oo as r —* oo such that

rH^{Ruf'°''tP{u) -> r € (0, oo) ;

then

P(X(f) <u, t<r) > e-'^, r—oo

where ip{x) = (2x)~-^/'^x~^e~*^/"^ and Ha is a

constant depending on a only.

(Here we use the definition of Ha given by Quedls

and Watanabe, Ref. [9], which is also used by

Cuzick and differs slightly from that given by

Pickands.)

The probability (1) is analyzed by splitting the

interval T into 'small' intervals T^, i = 0,1,...

and approximating the probability for an ex- ceedance of f{t) in T,. This suggests that only

the local behavior of the process is important

and that the assumption of stationarity might

be relaxed. Therefore S. M. Berman, Ref. [2], introduced the concept of local stationarity:

69
Definition 2 A real valued separable Gaussian

process {X{t),t £ T} is said to be locally stationary if

(i) EX(t) = 0 and EX^{t) = 1

(ii) There exist a continuous function R{t),t G

T with 0 < 'm{{R{t) : t e T} < sup{R{t) :

t E T} < oo and a strictly increasing con- tinuous function K{h),0 < h < ho {ho > 0)

with K{0) = 0 such that

lim

/i—

0

E{X{t+h)-X{t)f _ „2

K^{\h\)

uniformly in t £ T.

So the COvariance function of a locally stationary

Gaussian process has the form

p{t,t + h) = l- l/2R^{t)K\\h\) + o{K^{\h\)).

(2)

We will restrict ourselves to the case where

K^{h) is regulaxly varying at zero with index

a (0 < Q < 2). It can be shown that such a

process has continuous sample paths with probability one. Theorem 1 was generalized by J. Hiisler, Ref. [7], for locally stationary Gaussian processes and

nonconstant boundaries.

In Theorem 1 both the length of the time interval

(r) and the boundary u = Ur must tend to oo

in order to obtain a nondegenerate hmit for (1).

If r(< oo) is fixed and u oo or if r —> oo and the boundary is very high, then P(X(i) >

Ii, some t < t) —* Q and the question about its convergence rate arises.

Assume for the moment that X{-) is stationary.

Then by Theorem 1

P(X(t) > u, some t < r) ^ THc,{Rufl'^il}{u)

(3)

for large u (r < oo). Cuzick showed that (3) is indeed the correct convergence rate. He also obtained the convergence rate for the case, where

the boundary is a function of t. His result can be extended for locally stationary Gaussian processes in the following way:

Theorem 3 Let {X{t), t E T} be a separable locally stationary Gaussian process with co- variance function (2), where R^{-) is uniformly

continuous on T and K^{') is regularly varying

at zero with index a, 0 < a < 2 such that

K~^{-) exists in a neighborhood of zero. Let

{fn{t)y i G ^)n€]N <^ sequence of continuous

functions satisfying (fl)-(f3):

(fl)

(W

inf /„(0

teT

oo

where = K-\l/R{t)fM).

{fr.{t + rAr.m- Ut))fn{t) 9{t,r) , n—oo

uniformly in t E T and r in compact sets of

JR., where g{-,-) is a function satisfying

sup \g{t,T)\ < oo , V^, 0 < ^ < oo. teT

\r\<e

Then

-—P(X(i) > fr,{t), some teT) —, 1 n—»oo

with

A„ = X.«(.(.-))3gf.

-f$-(/„(0))-l(p(0,l)>0)

+*'(/n(r)) • l(r < oo and g{r,-l) > 0).

(4)

In (4) $*(x) denotes the tail probability of the

standard normal law and the functional Ha{w)

(0 < a < 2) is defined as follows:

Let {Xa{r), r > 0} be a Gaussian process

with Xa(0) = 0 a.s., EXacir) = -r''/2 and

70
Var(Xa(T) - Xaifi)) = |r - For continu- ous functions w[t), r > 0 define

where

= P (Xa(r) > s + \w{r)\, some r e [0, 6] n

with

L =

_ \ {0,a,2a,...}, a > 0

[0,oo), a = 0.

is defined as

H^{w) = lim sup H^{w, 6).

Let

Co = {w : w continuous and monotone on

[0,oc) and ^:j(0) = 0}.

The following Lemma was proved by J. Cuzick.

Lemma 4 IfwECo then

a)

uniformly in 0 < a < 1.

b) 0 < E-{w) <<x,,a>0 , H-iO) =

c) S^{w, 6) and H^{w) are jointly continuons

in a and w, where on Cq a sequence (lyn)nGlN

is said to converge to w iff

1) Wn{T) —> w{t) uniformly on compact

sets

2) (/o~e-l-'^WIdr)

-1

(/~e-l-Mlcir)

-1

The conditions on (/ti)tig]N imply that g{t,T) is linear in r, i.e. g{t,T) = C{t)T with some conti- nuous and bounded function C{t), t E T. Therefore H^{g{t^-)) is well defined, continuous in t and 0 < inft H^{g{t, )) < sup, H^{g{t, ))<oo.

For w(t) = ct (c € ffi.) one can show that

Hi{w) = 1/2 and H2{w) = (f>{c) - |c|$*(|c|).

Remarks:

- Cuzick's resiilt can be obtained as a special case of Theorem 3 by letting X{-) be sta- tionary with R{t) = 1.

- If X{-) is stationary with R{t) = R,

K^{h) = h^" and if r < oo, fn{t) = u^, then

A„ = rH^{Ru^)y^',p{u^).

- Note that Herman's condition is not necessary, since we are only interested in small

exceedance probabilities.

Let us consider two examples of sequences

(/n)Ti€]N which satisfy conditions (fl)-(f3). Suppose that K'^{-) has the special form K^{h) = h'^ and let r < oo. (f2) follows then from (fl).

Example 1: fn{i) '= + with /(•) positive

and continuously differentiable on [0,r]. Then

I 0. Q < 2

a = 2

Example 2: /^(t) := n^fit) {0 > 0) with

f[t) > (5 > 0 ,Vt < r and continuously differ- entiable. Then

0,

X),

Here (f3) is satisfied for a < 1 only

a < 1 a = 1

1 < a < 2

Sketch of the proof

As already mentioned, the excursion probability (1) is analyzed by splitting the interval T

71
into subintervals Ti = [ti,ti+i] of 'appropriate'

length. The split points are chosen as follows:

<o = 0

ti+l = ti + ^„An(ti), i > 1,

where ^„ —» oo 'slowly' such that for instance

^nA„(i) ^ 0 unifornily in t e T.

Let for i > 0

Ai = {X{tj) < U{ti), X{t) > U{t), some t G TJ,

where U = a,Tgmin{fn{U), fniU+i)}-

Then

P(X(t) > fr,{t), some t G T) = P(U(Wk)^/n(^i)>u^»))-

One can show that {X{ti) > fn{U)} (1 < ^ <

/) is either a subset of Ai^i or of At+i or its probability is of smaller order than P(A,) (7 =

sup{i > 1 : < r}). However, {(X(0) > /n(0)}

and {X{r) > fn{T^)} for r < oo are not negligible

if /n(0) < /n(ii) and /„(r) < /n(i/), respectively

(i.e. if p(0,l)> Oor 5(r,-l)> 0).

Thus

P(X(t) > some t e T)

« p(UAi)+$*(/„(0))- 1(^(0,1) >0)

+^*(/n(r))-l(5(r,-l)>0,r<(X)).

The proof consists of two major parts

1) Showing that

pu,)~/;*if„(,(,,.))3ggi...

This step is built upon the local behavior of

the process.

2) Showing that

t i Here the problem is to find a lower estimate.

To 1)

Suppose fn{ti) < fn{ti+i)- (The case /„(ii) >

fn{ti+i) is treated similarly.)

Write gr,{t,r) = (/„(f + rA„(i)) - /n(i))/n(0-

Using an idea of Pickands, Ref. [8], one can show

that

Jo

Jo IT ^^^"^^^^ '^^^MU, •))•

Application of the mean value theorem gives the

desired expression.

To 2)

Using Bonferroni's second inequality we get

t > ^p(^o(i-EP(^«n^)/EP(^«))-

Unfortunately, it is not possible to show directly

that the ratio term is asymptotically negligible.

Therefore one has to use a discrete time approximation first. The interval Ti is split into iV„

equally spaced intervals [^ij, 0 < j < N^,

where

Uj = U+ jCr^AniU) (an = en/Nn).

Writing = {X{U) < /„(li), X{tij) > uu,),

some 0 < j < Nn},

we have

i i

where

g„ = EP(^^nAj")/X;p(^°-).

72
One can show that

(some € > 0), which tends to 0 if 00 and

On —» 0 slowly enough. At this point the assumption (f2) about (/n)nelN is needed. In this part

of the proof the asymptotic linearity of fn{i) on

the intervals Ti is used several times.

As above one can show that

Since H^{g{t, •)) tends to Ha{g{t, •)) as a ^ 0

(Lemma 4), the proof is complete.

Application to first zeros of empirical

characteristic functions

Let y be a random variable with EIY'I'^ < 00

some P > 0 and write u{t) = Ecosty (real

part of the characteristic function) and cr^[t) =

Var cos ty. Let yi, . . . ,yTi be a random sample of

y and write Un{t) = {1 / n) cos tYj {real part

of the empirical characteristic function). Denote

by To and Rn the first zero of u{t) and Un{t)

respectively. Heathcote and Hiisler, Ref. [6], showed that if 1 — u{h) is regularly varying at zero with index q (0 < a < 2), then for r > 0

P(i2n < r) » P(X(t) > n}^^u{t)/a{t), some t < r),

where X{-) is a locally stationary Gaussian process with index a (covariance function p{t,t +

h) = l- 1/2(1 - Ti{h)){l/a^{t) + 0(1)) as /i -> 0). If a < 1 and r < tq, Theorem 3 can be used.

The CeLse a < 1 includes for example the Cauchy

distribution [a = 1), whereeis distributions with

finite expectation (a = 2) are not included.

Suppose y ~ Cauchy. Then u{t) = exp{—

i.e. ro = 00 and a^{t) = 1/2(1 - exp{-2|f|}).

Application of Theorem 3 yields for 0 < r < 00

P(i2n < r)

« (n/x)i/2 re-7(l-e-2^)3/2

Jo exp{-n/(e^^ — l)}dt as n —> cx)

/•CO = 1/(2^) / v-^/^e-^'dv

References

[1] Berman, S. M., Maxima and high level ex- cursions of stationary Gaussian processes,

Trans. Amer. Math. Soc, 160 (1971), 67-85.

[2] Berman, S. M., Sojourns and extremes of

Gaussian processes, Ann. Prob., 2 (1974),

999-1026.

[3] Berman, S. M., Sojourns and extremes

of stochastic processes, Wadsworth and

Brooks, Pacific Grove, 1992.

[4] Braker, H. U., High boundary excursion of

locally stationary Gaussian processes, Ph.D.

thesis, University of Berne, 1993.

[5] Cuzick, J., Boundary crossing probabilities

for stationary Gaussian processes and Brownian motion, Trans. Amer. Math. Soc, 263

(1981), 469-492.

[6] Heathcote, C. R. and Hiisler, J., Estimating

the first zero of an empirical characteristic

function, Stoch. Proc. Appl., 35 (1990), 347-

360.

[7] Hiisler, J., Extreme values and high boundary crossings of locally stationary Gaussian

processes, Ann. Prob., 18 (1990), 1141-1158.

[8] Pickands, J. IH, Upcrossing probabilities

for stationary Gaussian processes, Trans.

Amer. Math. Soc, 145 (1969), 51-73.

[9] Quails, C. and Watanabe, H., Asymptotic properties of Gaussian processes, Ann.

Math. Stat., 43 (1972), 580-596.

73

Asymptotic Approximations For The Crossing

Rates Of Poisson Square Waves

Breitung, K.

Sem. F.A. Stochastik, Akademiestr. 1/IV, Munich, Germany

For the extreme value distribution of functions of random vector processes it is difficult to derive exact expresssions; therefore approximations are needed. A mdel commonly used for loads in struc- tural reliability is the Poisson square wave processes x{t). Such a process is defined by a Poisson

point process and an additional sequence of i.i.d. random variablejs Vo. ^i) • such that between

two points tn and tn^i of the Poisson process the value of x{t) is defined by x{t) = Y^. In this paper

the asymptotic Poissonian behavior of the point process of level crossings of functions of independent Poisson square wave processes is shown. This can be used to approximate the extreme value

distribution of such functions.

1 Introduction

In many reliability problems it is necessary to calculate the distribution of the maximum of functions of vector random processes. A survey of random processes used

ai, models in load combination problems is given in [1]. For many models it is difficult or impossible to derive

the exact distribution of the maiximum. Therefore it

is of interest to obtain asymptotic approximations for high levels, since especially such results are needed in reliability problems.

Here for a special model, Poisson square wave processes, the asymptotic Poissonian behavior of the point

process of level crossings of functions of such processes

will bo described.

x{f.)

Yo Y3

Y2

t2 '3 t Figure 1: Poisson square wave process

A Poisson square wave process consists of an homogeneous Poisson point process N{A) with intensity A and a sequence Yo,Yi, Y->, ... of i.i.d. random variables,

which are independent of the point process.

'I1ie Fois.sori square wave process x{l) is then defined

by

x{t) = Yr,^o,t]^Y,, i{X{0,t] = j. (1)

The process is often used lo describe loads wich

change in time (see [2], [1] and [3]). The model can

be generalized to a vector process by taking random

vectors in.stead of random variables. Tn none of the ref- erences above a complete proof for the asymptotic Poissonian character of the level crossing point processes is given.

We consider now an n-dimensional Poisson square wave proce.ss x{t) — {xi(t) Xn{t)). The i-th component is defined by an one-dimensional Poisson square wave process with intensity Aj and standard nor- mally distributed amplitudes. All component processes

are assumed to be independent of each other. The process x{t) changes its value at the jumps of the component processes Xi(t). As described for example in [4] and [1], p. 73, all random variables X with c.d.f. F{x)

and with a continuous p.d.f. f[x) can be transformed

75
into standard norma] random variables U by the transfornialiofi U - <p-\F{X)) (<&(x) the standard normal

integral); therefore the assumption is not too restrictive

in the sense that we need this transformation only to derive a simple proof for Poisson convergence. Further a function g : JRP' —* R is given. In the

reliability context, this function describes the state of

the engineering system under consideration. If g{x) >

0, the system is intact and if g{x) < 0, the system is defect. The problem is to determine the probability

PTigixit))>0, 0<t<T). (2)

This is the probability that the system remains intact

during the time interval [0,7']. The simplest example

for such a function is the sum of the processes, i.e. g{x{t)) = 0- Y:"=i Xi{t) with peR.

To find approximations for this probability, we consider the point process of the outcrossings out of the

domain S = {x;g{x) > 0} into the domain F =

{x;g{x) < 0}. The boundary of F is defined by

G={x;g{x) = 0}.

Firstly, the point process of jumps o{ x{t) is defined

l^i^A) = #lteA; A jump of one of the com- 1

^ [ ponent processes Xi{t) at t. J Then, the point process of outcrossings U{A) out of

S into F is defined by

U{A) = #{t e A;gixit - 0)) > 0 > (3)

Now, since x{t) changes its value at the jump times

only, we obtain

1 - PT{g{x{t)) >0\0<t <r)

= l-PT{g{xm>0,UiO,T) = 0) < Pr(5(x(0)) < 0) -f Pr(C/(0,T) > 0).

(4)

Since ^''(0, T) is a non-negative random variable, we

get

1 - Pr(<7(x(0) > 0;0< < <T') < Pr(ff(x(0)) < 0) + E{U(0,T)).

(5)

Therefore, we obtain an upper bound for the probability in equation 2. In the following we show that then

under some regularity conditions, U{A) is asymptotically an homogeneous Poisson process. This gives

Pr(f/(0,T) = 0) ^

with Af/ the intensity of the point process U{A).

Then wc got

1 - Pr(y(x(0) > 0;0 < i < T)

«Pr(y(x(0))<0) + (l-e-^'^-^).

(6)

(7)

2 The asymptotic behavior of

the outcrossing point process

Consider a Poisson square wave vector process x{t) as defined in the last paragraph. Given is further a continuos limit state function g{x) with niin5(x)=o |*| = 1 and there is a unique point xq on G with |a;o| = 1 such

that near this point g is twice continously diffcrcntiablc. This means that the function |x| has a minimum with

respect to 0 al xq- We assume that this minimum is regular, i.e. the main curvatures ki, . .

. , «„_! of G at xq are less than unity. (This follows the differential geometry of G, see for example [5], chap. 12.) We define two sequences of domauns Sp =

{x-gip-^x) > 0} and Fp = {x;g{/3-^x) < 0}. Now

we consider the point processes Up{A) of outcrossings

of the process x{t) out of Sp into Fp, defined by

Up{A) = tit e A; gp{x{t - 0)) > 0 > g{x{t))]t. (8)

We will study the asymptotic behavior of the stan- dardized point processes Up{A), defined by

U'p{A) = Up{E^' A), with Ep = E{Up{0, 1)). (9)

These processes converge towards a Poisson point process under some regularity conditions.

It will be assumed that

n n xo = Y^aiCi, ^cij = 1, > 0, 7 = 1, . . .,n. (10)

1=1 1=1

Here is the unit vector in the direction of the n-th

component. This means that all direction cosines of

are not zero. We define

ao = mm |a,|.

i = l,....n (11)

Now, due to the definition of xq, there exists a ($ > 0

and an < > 0 such that for all y with g{y) < 0 and

IZr=i Vi'^i — y'^^o < (1 — ^)!y| (i e. the cosine of the

angle between y and xo is less than 1 — ^)

\y\ > (i + €)ix o|- (12)

Elsewhere there would be another point on the surface G = {x;g{x) = 0} with unit distance to the origin. Let be defined

Fp = {x;(7Hx)<0.x^xo<(l-(^)|x|} (13)

F; = Fp\Fp. (M)

see figure 2. The number of the points of the outcrossing point

process Up{A) is bounded from above by the number

76
we get asymptotically

Figure 2: The sets and Fp.

of the points of the point process Up (A), which counts

all jumps of the process x{t) into a point in F/?

^7^(.4) = #{t € A; x{t - 0) ^ x(t), gp{x{t)) < 0}#, (15)

i.e. Udi-A) > Ufi{A). Further, for the point processes

U0{A) = #{t e A-g0{x{t - 0)) > 0,x{t) G Fp}#, (16) we see by bounding from above that

E{Up{A)) (17)

< |.4i P'i9ft{x{t - 0)) > 0, x{t) e /»

< \A\[Y^xAPT{x{t)^F0)

< 1.41(5;^ A. )Pr(|x(OI>/?(l + 0) = o($(-/?)),/?-^.x). (18)

The last inequality follows, since |a!(OI^ ha.? a x^-

dislribution with n degrees of freedom and using equation 26.4.5 in [6] we get Pr(|a(OI > P{\ + f)) -

(J(l + 0)""-exp(-;3(l + e)) = o(<^(-/?)). Therefore E{U0iA) - UffiA)) = o{EiU0iA)), 0 — oo, i.e. asymptotically the point process of jumps which [cads

to points in F^ is negligible in comparison with the

point process Op{A).

For the point processes UgiA) we get

EiUpiO, 1)) = • PK.<7;3(x(0) < 0). (19)

Using the results of [7] and [8] about asymptotic ap- proximations for the probability content of domains.

n- 1 Hr(g/5(x) < 0) ~ <t>{-f:)) ^(1 - H - <x>, (20)

and therefore

^(^7^(0,1)) (21)

Since f/^(A) > [/^{A), this yields an upper bound for E{U0{O,l)).

We show that this upper bound is in fact an asymptotic approximation for this expected value as /? ^

CO. This is done by proving that the expected value

E{U0{0, 1) — (7/3(0, 1)) is asymptotically negligible as

/? —» oo. The point process UpiA) — (/^{A) consists of

all points t of Up{A) with g0{x{t - 0)) < 0, i.e. the

points where the process was immediately before the

jump in F/3 and afterwards again in this domain.

The probability of obtaining a point in the domain Fp

after a jump is of order o(<I>(— /?)), as shown in equation

17. If the jump results in a point in the domain F^, it

is obvous that this is due to the occurrence of a new

component amplitude, which has a value larger ttiati ao /?/2, since F^ C {x; mini-i,...,„ |x,| > ao • /?/2}.

Therefore the expected number E{Up{0, 1) - Up{0, 1))

is less than the expected number of jumps multiplied

by the probability PT{gp{xit)) < 0) and the probability

of a new component amplitude larger than anfi/^.

An upper bound for the first probability is one and

the last probability is ^>(— qq J/2). Therefore

E{Up{^.l)-Up{QA)) (22)

< {fl^^ rr(5,(x(?-0)) < 0)

x[<I>(-ao/?/2) + o(^(-/?))].

Asymptotically we get usmg equation 20 for the ex- pected value

E{U^{[).\)-Up{i)^)) cm

- (^|:A.j<&(-5)na-«;)-^/^

- o(<J>( -.?)),/? — oo.

This shows the asymptotic equivalence of the two point

processes and gives linally

lh:[Up{i),\)) (24)

n-1

77
Tf there is instead of one minimal distance point Xq

a finite number of points xi , . . .

, with regular minimal distance points, an analogous result is obtained by

splitting up the domain F into k disjoint sets F\, . . . , Fk

with uf_,F, — F such that for each F,- exactly one

point X, lies on the boundary of F,, by treating each

set scpcratcly and then adding the results. Wc get then

m=l ] = l

, /? — OO.

with the KmJthe main curvatures of the surface G at

We prove the convergence to a Poisson process for the

standardized processes U^{A) in the next paragraph.

To this purpose we replace Ui3{A) by an approxiiiiattiig

point process

Ui,{A) = #{t e A;g{x{t)) < 0, \x{t-0)\ < /?- log(/:^)}#

This is the point process of all outcrossings from Sp

into Fp, whicli start from a point in the sphere around

the origin with radius (3 — log(/3); since this sphere is inside Sp, we have Up{A) < Up{A) (see figure 3).

circle radius P

circle radius (1-logp

Figure 3: Approximating point process Ffl{A)

Using similar arguments as before it can be shown

that the two point processes Up{A) and Up{A) arc asymptotically equivalent. Therefore, if Up{A), i.e. in standardized form, converges to a Poisson process as J — oo, this is valid for U^{A) and U^{A) too. In the next paragraph it is shown that for such a conver- gence can be proved.

3 Convergence to a Poisson process

To show the convergence of the standardized processes U^{A) to a homogeneous Poisson process, the following

result is used. Given is a sequence Np{A), 0 > I of stationary and

orderly point processes with the following properties:

1. Fp = E{Np{0,l))-^0,

2. There exists a function a : (0,oo) —> El with

a(r) — 0 as r —H- DO such that for all events Ap

and Bp, where Ap depends only on the behavior

of the point process Np{A) until the time t and

Bp only on the behavior of Np{A) after the time

I PiiAp n Bp) - PxiAp) ?r{Bp)\ < a(rF;^). (25)

i.e. the standardized processes A^^(^) are uni- formly mixing with mixing coefficient a(r).

p-l/2 p-l/i

3. / / ppiti,t2)dt: dt2 = o{Ey^), 0 ^ <x>, 0 0 where pp{t\,t2) denotes the two-dimensional product density of the point processes Np{A).

Under the conditions 1 — 3 above the .standardized processes Np{A) = Np{Ep^ A) converge to an homogeneous Poisson point process with intensity 1. As outlined in [9], p. 37 for a stationary and

orderly point process Np{A) the factorial moment

E{Np{0,t)){Np{0,t) - 1)) is given by

EiNp{0,tmNB{OJ)-l)) (26)

ti.t2)dti dt2

0 0

t t2 = '^- J J

P0ih-U)dti dt2.

0 0

This result can be shown by using theorem 1.3 in a paper of Volkon.skii and Ronzanov ([10]). This the- orem says that stationary and orderly point processes converge to a homogeneous Poisson point process, if the conditions 1 and 2 are fulfilled and further for i —' 0 and /? — oo always

PT{Np{0,Ep -t) > 0) ^0. (27)

But this condition can be replaced by the condition

given in equation 1.38 in the paper of Volkonskii and

Rozanov, i.e.

JE{Np{0, Rpt))/E{^p{0, Fpf.)-^], 0^cx>. (28)

78
Due to the stationary of the processes it is sufficient to — 1/2 show this for the intervals (0, A'^ ). The third condition above says that

E [n,AQ, E;"\Ns{^, E-'") - 1)) (29)

and therefore, since we have

Integrating over this function gives

E{N^{0, E^''''))/E{N0{O, E-"')) ^ l, /? -> c«. (30)

Therefore then relation 28 holds. So the conditions 1-3 are sufficient for convergence to a Poisson process. We show that the processes U0{A) fulfill the conditions; from this follows the result then for U(j{A)

too. To prove this we note that equation 24 shows

that the first condition is fulfilled, since Ep =

(Er=i nr=i'(i - •^^r"'' - o as /? c^. The second condition is fulfilled, if we take as strong

mixing function Qr(r) = Yll=\ This' follows from

the fact that the underlying point processes Ni{A) are homogeneous Poisson point processes with intensities

\i and therefore Pr(7V, (0, t) = Q) = e-^•^

To show the third condition, we split the joint intensity function, denoted by fp{ti,t-i), into two functions:

<2) (31)

with

-1/2,

Pr

Pi{t2-h) (32)

All point processes A^,-, i = \, . . . ,n have at least one point in (ti,^?)-

=/'(«t-tj)

P2(<2 -h)=\- Pl(«2 - h) = Pr(/(<i - t-iY).

Further fp{ti,t2) (rcsp. /^(<i,<2)) denotes the conditional density of the point process Up{A) under condition I(t2 — ti) (resp. under condition /(to — ^i)*^)- In the first case all point processes Ni, i = l,...,n

have at least one point between the two time points ti and l2- The probability for this event is P{i2 — ^i) =

i - n?=i Pr(A:(^2, ti) = 0) = l- riLi e-^'('^-'') < 1. The condition means that all loads have changed and

that therefore the behavior of the point processes Up{A)

at the points ti and t2 is independent under this condition. Therefore the conditional joint intensity /^(<i,<2)

is just the product of the constant one-dimensional in- tensity Xff of the process at these points. Since the process asymptotically equivalent to the process (/p{A), wc

get from equation 24 ^ Ep iis —< oo. 'I'lierefore, we have

2- j I

Px{h-li)fl{h,l2)dh dt2 (34)

b b 1/2

<2- y j{El + o{El))dli dl2 = o{El'^~), 0 -^oo.

0 0

In the second case, at least one of the point processes Ni had no point in the time interval between ti and t2 and therefore at least one of the amplitudes did not

change. Since these processes are independent Poisson

point processes, the probability for this is bounded by

o<P2(i2-^i)<Er=i^~''^'^"''^-

The conditional intensity in this case is less than the

intensity of the point process of occurrences of a component larger than log(/?) at t2, since only in this case a jump of the point process Up{A) will be at <2, multiplied by the one-dimensional intensity of Up{A); this intensity is asymptotically equal to Ep.

Therefore, as /? —>• oo

P2{t2-h)fl{tut2) m

Integrating over this function gives

- j j

P2{t2-ti)fpiti,h)dti dt2

0

<2(f^A.]<I>(-log(/^))2^;3

0

'0 t2

PAh-tx)f'pituh)<E^fl + oiE}). (33)

0 0 = KoiEf'M- log(.5)) + o(£y-)) = o{E'/'), 13

with Kq a constant. Together with the upper bound for the first term in ecjuation 34 this yields

E{Up{0, E]'^'){Up(0, E;"-) - ])) (30)

= 2 J j

fp{tu h)dtx dt. = o{E\'^^), (j-*oo.

0 0 So the third condition holds too. Therefore we have for the standardized point process the convergence to a Poisson point process, i.e.

Pr(C/j(/l)=rO)~c-l-^l, /?-*oo. ' (37)

79
4 Summary

In this paper a Poisson convergence theorem for the

point process of level crossings of functions of independent Poisson square wave processes is shown. The idea

is to split the two-dimensional joint intensity of the

point process into two conditional intensities, one de- scribing the behavior under independence of the processes at the two time points and one under dependence. Then by estimating it from above, it is shown

that the second intensity can be neglected asymptotically; this gives the Poisson convergence.

5 Acknowledgement

The main results of this paper are based on re- search done in a project supported by the Deutsche

Forschungsgemeinschaft

.

References

[1] n. Madsen, S. Krenk, and N.C. Lind. Methods of

Structural Safety. Prentice-Hall Inc., Englewood

Cliffs, N.J., 1986.

[2] K. Breitung und R. Rackwitz. Nonlinear combination of load processes. Journal of Structural Mechanics, 10(2): 145-166, 1982.

[3] K. Schrupp and R. Rackwitz. Outcrossing rates of

marked Poisson cluster processes in structural re- liabiUty. Applied Mathematical Modelling, 12:482-

490, 1988.

[4] M. Hohenbichler and R. Rackwitz. Non-normal

dependent vectors in structural safety. Journal of the Engineering Mechanics Division ASCE,

107(6):1227-1241, 1981.

[5] J. A. Thorpe. Elementary Topics in Differential Geometry. Springer, New York, 1979.

[6] M. Abramowitz and I. A. Stegun. Handbook of

Mathematical Functions. Dover, New York, 1965.

[7] K. Rreitimg. Asymptotic approximations for multinormal integrals. Journal of the Engineer ing

Mechanics Division ASCE, 110(3):357-366, 1984.

[8] K. Breitung and M. Hohenbichler. Asymptotic ap- proximations for multivariate integrals with an application to multinormal probabilities. Journal of

Multivariate Analysis, 30:80-97, 1989.

[9] D.R. Cox and V. Isham. Point Processes. Chap- man and Hall, London, 1980.

[10] V.A. Volkonskii and Yu.A. Rozanov. Some li theorems for random functions. I. Theory of P

ability and its Applications, 4(2): 178-197, 1959

80
Meso-Scale Estimation Of Expected Extreme

Values

Burton, R.M., Goulet, M.R., and Yim, S.C.S.

Oregon State University, Corvallis, OR

We consider algorithms for estimating the expected maximum value of a time series for a

period in the future given past observations. This is a "mid-range" problem in which the long term

cisymptotics of extreme value theory do not apply. There are essentially two approaches, estimating an "extremal index" and the "Poisson clumping heuristic". Variations on these methods are tested

with simulated Gaussian data. Similarities in performance are explained rigorously.

INTRODUCTION

We consider the following problem. Given a time

series from a stationary process {Xi}'^-^, define the ex- pected maximum E[Mn,N'], where N' > N and

Mn N' = max X-

.

N+l<i<N'

The problem is to find a good estimator of E[M^r J^;] based on observations {Xi}\L-^. We will alway consider

Gaussian time series but it will be clear that our methods apply more generally.

Here we describe several estimators of E[Mn,n']-,

present some empirical results and give some theoretical explanations of our results. DESCRIPTION OF THE ESTIMATORS

Time Rescaling. The idea is to estimate an ex- tremal index of the process for this time scale. We say

that p is the extremal index for {Xi]f2.-^ on the scale of N if M^- has approximately the same distribution

as the meiximumof [pN] independent random variables

with the same distribution as Xi, i.e Gaussian.

There are essentially three choices to be made in approach.

The first is whether to use the time series itself or an enveloped version of it. Given the data, {Xi}fLi we may construct the discrete Hilbert transform {Yi}fLi.

The process {Ri}iLi defined by

R, = yjxf + Y,'

is called the analytic envelope. It covers the "surface" of the time series, smoothing out the oscillations. The

maximum of the envelope is close to that of the original process, especially in the narrow band case. It has

the further computational advantage of being Rayleigh

distributed. This is described in detail in Ref. [1, 2]. We call these choices direct and enveloped.

The second choice is how carefully to compute the

expected maximum of n independent Gaussian ran- dom variables (or Rayleigh in the enveloped case) as a function of n. One could either use a good but computationally intensive numerical approximation or an

asymptotic formula, a/2 logn. We will refer to these

choices as strong and weak and call this approximation we use L(n).

The third choice is how carefully to fit the empirical

expected maxima as computed from the data to L{n).

One possibility is to use one value of no, say 50. Find

the average of the maximum value in the data for nonoverlapping windows of length no. This value L(no) is the empirical expected maximum at no. To estimate

the extremal index then find p so that Z(no) = L{pno).

The other possibility is to compute multiple window

lengths, i.e. to compute L(n) = L{n) for various n. If we take n to be powers of 2 then the computation time

is not large because we may "nest" the computations

of the maxima. We will refer to these choices as single

window and multiple window methods.

We note that the prevailing method among ocean

engineers was the enveloped, strong, single window

method.

81
Poisson Clumping. Another possible estimator

is suggested by Aldous' use of the Poisson clumping

heuristic, Ref. [3]. This heuristic assumes that the

set of t for which Xt > b is given by random sets dis- tributed as a Poisson process. We make the further

assumption that these random sets are intervals. Consider, for b relatively large

{t\Xt.^<b,Xt>b}

to be distributed as a Poisson process with rate Aj. The

following fundamental relation is assumed

P[Xi >b] = XbE[ai

where Cb is the random length of an interval (clump)

in which the time series spends above a given value 6. The event [Mn < b] is equivalent to

{t\Xt-i <b,Xt>b} = 0.

So by the Poisson assumption,

P[Mn <b] = e-^>'^,

and by the fundamental identity

Hence we have

E[Mn] = /"(I - e-^[^'^*l^/^t^'])rf6. (1) Jo

The work now reduces to estimating E[Cb] - To do this we fix a value of 6 and average the length of the in- tervals where the time series is above b. Varying b and

plotting E[Cb] versus 6 yields data which is well fit by

a curve of the form

y = b-yA.

Substituting this curve into (1) yields our estimator,

Ep[Mn,n'] = / (1 - e-^^[^^^*K^'-^)*')d6.

Jo

As before we may use either the original or en- veloped data. Note that the above analysis assumes

that the clumps are intervals so one guesses that en- veloping narrow band data would be advantageous.

EMPIRICAL RESULTS

We have described 10 possible algorithms in all. In earlier work, Ref. [2], we investigated several al- gorithms. The algorithm used by most ocean engineers was due to Pierce, Ref. [1]. In our terminology this was an enveloped, strong, single window time rescaling

method.

We proposed to modify this by removing the envelope, that is to use instead the direct, strong single

window time rescaling method. In cases where computation ease Wcls paramount we proposed the direct,

weak, multiple window rescaling method. These were compared with the direct Poisson clumping algorithm.

Here we describe the results of these simulations. In subsequent versions of this paper we will also include

study of the enveloped Poisson clumping algorithm.

Work continues on the other variations. In this study two types of Gaussian time series are used. The first is a second order autoregressive moving

average. Xn = aXn-1 + bXn-2 + Zn,

where the Z„ is are independent identically distributed

Gaussian random variables.

The second type is intended to simulate random

waves in the ocean and are obtained by superposition

of sinusoids, with amplitudes specified by the PiersonMoskowitz and JONSWAP spectrums, Ref. [4]. One

thousand cosines with unequal frequency spacings and

uniformly random phases are employed. More detail on these processes is given in Ref. [2]. These time series are run for various parameters

and the expected maximum are estimated by the al- gorithms. The mean relative errors are computed.

0.000 0.200 0.400 0.600 0.800 1.000 1.200 Figure 1

Our results are pictured in Figures 1-3. To summarize the ARMA experiments (Figure 1), the Direct

Method consistently gives the estimator with minimal

relative error, while the Poisson clumping and Log Fit

methods yield estimators with relative errors under 6%.

It is interesting to note that the results for Poisson

Clumping and the direct, strong, single window time

rescaling method follow each other.

82
References.

0.090 -

0.080 -

0.07D -

0.060 -

O.0SO -

O.OtO -

0030 ;-

0.020 -

0.010 -

0.000 -

-0.010 -

0.080 |~

0.070 |-

0.060 -

0050 -

0.040 -

0.030 -

0.020 -

0.010 -

0.000 -

-0.010

Figure 2

birca

'lo( Fit

[1] R.D. Pierce. Extreme value estimates for arbitrary

bandwidth gaussian processes using the analytic en- velope. Ocean Engineering, 12(6):493, 1985.

[2] R.M. Burton S.CS. Yim and M.R. Goulet. Practical methods of extreme value estimation based on measured time series for ocean systems. Ocean Engineering, 19(3):219-238, 1992.

[3] D. Aldous. Probability Approximations via the Poisson Clumping Heuristic. Springer-Verlag, New

York, 1989.

[4] T. Sarpkaya and M. Isaacson. Mechanics of Wave

Forces on Offshore Structures. Van Nostrand Reinhold, New York, 1981.

[5] G.L. O'Brien. Extreme values for stationary and

markov sequences. Annals of Probability, 15:281-

291, 1987.

[6] H. Rootzen. Maxima and exceedences of stationary

markov chains. Advances in Applied Probability,

20:371-390, 1988.

Figure 3

For the simulated ocean waves the results are similar. In the case of the Pierson-Moskowitz spectrum

(Figure 2) the Direct Method consistently provides the

best estimator, regardless of the dominant wave length.

The JONSWAP spectrum (Figure 3) provides a narrow band case and all the techniques yield relative errors between 4% and 7%, while the Log Fit provides the

best estimator in two instances.

CONCLUSION

The theoretical underpinning for these algorithms is given by theorems of O'Brien, Ref. [5], and independently by Rootzen, Ref. [6]. There it is shown for processes satisfying a strong mixing condition (as ours do)

that for long enough time scales there is an extremal

index which in turn gives the Poisson clump structure

of the exceedance process. Thus it is not surprising to

find the direct, strong, single window estimator and the

direct Poisson clumping estimator in close agreement.

Perhaps the most striking result in this study is the

performance of the direct, weak, multiple window esti- mator. It is the simplest conceptually and algorithmically, and gives relative errors near 6%.

83

An Expert System Prototype For The Analysis

Of Extreme Value Problems

Castillo, E., Alvarez, E., Cobo, A. and Herrero, M.T.

University of Cantabria, Santander, Spain

This paper presents an expert system prototype for the analysis of extreme value problems.

The system includes a package of computer aided instruction for the most common concepts

in Extreme Value Theory and illustrative examples of applications. The user can navigate

through the information at wish. The system incorporates a computer program to simulate,

estimate, draw samples on extreme probability papers and determine the domain of attraction

of a parent from samples, based on the Pickands' and/or the curvature methods. A set of rules

controls the selection of probability papers and estimation methods adequate to given

problems.

Key words: Extreme value problems, expert system, simulation, estimation.

1 Introduction

Extreme value problems are very frequent to engineers. In fact, in many engineering situations,

design is based on the probability of occurence of

extreme values of single or combined random

variables. When dealing with extreme value

problems, engineers and scientists find some

difficulties due to the following facts:

• Extreme value theory is complicated.

• Extreme value theory is not easily available.

• In general, technicians have not been

prepared to deal with this problem.

In spite of its importance, extreme value theory has

not been included in standard curricula. Even in some specific fields, such as statistics, for example, a very large part of the student population ignore

fundamental aspects of this theory. This problem is even more important in engineering areas where a

large amount of technical material must be covered.

On the other hand, extreme value theory is not

easily available to those who need it. Most of the

advances have been published in journals and books

mainly addressed to probability and statistic

specialists and using a langauge difficult to understand for engineers and scientists.

The consequence of all the above is that important

errors and inconsistencies have been made in the

past, such as: the use of incorrect models or

probability papers, the use of non-stable models, the use of wrong estimation methods, etc. It is easy to find examples of limit models for minima that are used for maxima and vice versa, or examples of

incorrect use of probability papers. In other cases, non-stable models, either in extreme or truncation

operations, lead to confusions and lack of

consistency.

All the above justifies the need for expert systems.

We must remind the reader that expert systems are useful mainly when (see Ref[l]):

• there is a lack of human experts

• there is lack of knowledge among those

who need it • one needs more reliable solutions

• one needs cost reduction.

To our knowledge no expert system exists covering

all these needs. This paper addresses this problem

and presents a simple prototype showing some of

the excellences an expert system should have.

85
2 Steps in the development of an extreme

value expert system

Expert system design must be carefully programmed if success is desired. Some of the main steps to be followed are (Ref[2]):

1 . Statement of the problem to be solved.

2. Searching for human experts and/or data.

3. Design of the expert system.

4. Selection of the development tool, shell or programming language.

5. Development of a prototype.

6. Prototype checking.

7. Refinement and generalization. Final expert

system.

8. Maintenance.

9. Updating.

The first step consists of defining the problem to be

solved. Under no circunstances should time spent on

this period be curtailed, and work should be done

with rigor and precision. All extra time dedicated to

this step will be saved in the following steps. This

step implies identifying all or a large part of the

possible difficulties to be encountered when dealing

with extreme values and the possible solutions to be

employed. When this is clear, a decision about which

of them are to be solved by the expert system must

be taken.

Once the problem has been completely defined, one must look for human experts able to solve it with a reasonable chance of success.

The third step is the design of the expert system,

which includes the structures for knowledge storage,

the inference engine, the explanation subsystem, the user interface and so on.

In the following step we must decide whether to use

a shell or a programming language. It is important

to avoid useless efforts that are also expensive. The

final steps cover the prototype development,

checking, refinement and updating.

3 Minimal requirements

In this section we discuss some of its minimal

requirements. The expert system should include at

least: . ,

, -

• Computer aided instruction on extremes.

• Tools for doing statistics of extremes.

• Tools for gaining experience and expertise.

• Bibliographic information.

The first part should cover the most important

concepts relevant to extreme values by means of:

theory, illustrative examples of applications,

interactive methods, such as, animations, examples,

hypertext (user driven navigation through

information), hypermedia, guided tours, etc.

The new computer aided instruction techniques,

based on hypertext and hypermedia, allow an easy

and quick development of a teaching module. A

guided tour guarantees the most important concepts

to be covered and aprehended by the user.

Determination of design values or probability

assessments are the result of an iterative method

based on a combination of different steps as drawing

data, selecting models, estimating parameters, etc. Thus, the tools for doing statistics of extremes

should cover:

• Drawing data on probability papers and

other graphic representations.

• Model selection.

• Estimation.

• Determination of domains of attraction. • Determination of design values.

Another interesting role of an expert system is its possible contribution to the user in gaining

experience and expertise. For this to be possible, the

system should include:

• Access to real cases (data base).

• Simulation.

The easy access to previous experience and methods

facilitates the solution of many real problems. The

system could include practical cases and different

alternative solutions given to some typical problems.

A simulation system allows for gaining experience

and testing the appropriateness of some temptative

methods. Note that testing of several alternatives can

lead to a very useful information thus facilitating the

final decision.

Finally, a complete bibliographic information

including cross references should be available. This

facilitates the access and navigation of the interested

reader through the information.

4 Prototype description

The implementation of an expert system for the

analysis of extremal problems is a complicated task,

which must involve a group of people. In this section we present a simple prototype including only

some of the above possibilities to show the

convenience of such a system.

86
We shall divide the exposition in three parts: the

computer aided instruction subsystem, the statistical tools subsystem and the proper expert system.

4.1 Computer aided instruction subsystem

This package covers the material indicated on the

menu card (see figure 1), where different options can be selected.

The user can select one of the topics by just clicking

the mouse on it and the system shows the relevant

information associated with it. In some cases this produces a new menu, as in the case of figure 2,

where new options can be selected.

With the purpose of introducing some concepts some animations are used, such as those illustrated

in figure 3, where samples are drawn at random

from the population and in figure 4, where the

whole process of simulating order statistics is animated.

In other cases, graphical information is used to

illustrate concepts, such as that indicated in figure 5, where the curvature of tails is used to decide about

domains of attractions. Other cards are used for

definitions of relevant concepts (see figure 6).

Information is structured in such a way that the user can go back and forth at wish, consulting examples,

definitions, graphics, animations, etc.

4.2. Statistical tools subsystem

The expert system incorporates a set of computer

programs to do a statistical analysis of data.

If we choose reading data from a file, a dialog with

all available files is shown (see figure 7). Then, the

file is opened and data appears in the text window.

Once the statistical program is launched, a new

document is opened and two empty windows on the

computer screen are obtained. They hold the sample

data and its associated drawing, respectively.

Figure 1 : The main menu.

87
ORDER

;

STATI

1

^—. T •> ^ iL-a Cj STICS- ;

Dertnitlon of order statistic

Joint distribution of several order statistics

Distribution of a single order statistic

Order statistics in samples of random size

Simulation of order statistics

lllil Click here to get some information

Figure 2: The order statistics menu.

This Is the ordered sample

(1 4,1 a,24,29,30,35,38,39)

M the statistic of order 8 is ; 39

'^'V'^ ~'?^w'>i. -'*"!^

Figure 3: The concept of order statistic.

88
Figure 4: Simulation of order statistics.

EKtremes

The curv^ure Caachy distribution saggesU a^Frecliet'' type iommn of flitFatcticn The curv^iB-e of Normid di^ribiiHon sugsests a 'Gumbe]' type demadntiof ,«ttr««U9n

The curvaJture <^ Un^onn distrifoution suggesits a'^eibuir type domain of tttraction The curvatis-e of Exponentnd dts^tribation suggests a'^Swabel'' type domain of attraction

Figure 5: Determining maximal domains of attraction.

89
tfs correspond to the selected pk>ttln9

posttfon f<WTOula and F(x;8) is th^ ftimlly of distributions being

, fitted. Note that each terrn nieasures the refatrye eroor In redjrn

I period, thus justifying the name of the method.

|- It IS iobwesting to point out thet this method coirtcides with the

;\tail equfyalence method. We say that two d^trlbutiof» F(x) and

1 13(x) are tali equivalent we have: = 1

Usirtg thi criterlum we can minimize the ssi^ of squares:

'5 = Z -[pi-F(S(i);8)f

Figure 6: Least squares methods.

The program allows for the four operations shown

in the main menu of figure 7.

Initially, due to the lack of data, only the "simulate"

option is available and the rest appear as dimmed.

At this step we can choose either reading data from

an external file, typing data directly on the text window or simulate data.

If we choose the "simulate" option, the dialog in figure 8 appears to allow us to choose the

distribution, its corresponding parameters, sample

sizes and range of order statistics to be simulated. In any case, the text window ends up with the working

data. At this step all the options above become

available.

Selection of the "draw" option of the main menu

leads to the dialogs in figures 9 and 10, where the

desired probability paper and plotting point position

formula are selected. Then, the drawing of the

indicated probability paper and the sample is obtained (see figure 11).

Selection of the "estimate" option leads to the

dialogs in figures 12 and 13, where the method of

estimation and the family of distributions to be used

in the fitting procedure are selected. Then, the range

of order statistics is given and the system initiates the

estimation process and informs the user of its progress. At the end, it gives the estimates and, in some cases, the variance-covariance matrix of

estimates as is shown in figure 14.

^

EKtremes c=i Macintosh HD

1 D Loads

I ) 1 liiaues LUinds [ Desktop ]

f Cancel ]

l( Open )|

Options

Simulate...

Draui... Estimate...

Domain of attraction.

Figure 7: Reading from a file and the options menu.

90
Simulation

Choose one distribution: r— Distributions

<i) Gumbel (maKima)

O Gumbel (minima)

O lUeibull (mawima)

O Uieibull (minima)

O EKponential

O Uniform

O Cauchy

O Rayleigh

O Frechet (maKima) O Pareto O Frechet (minima) O F(K)=eKpI-1 /(k*k)J

Sample size:

Ca

Uihole sample

ncel ) [[ OK \

Figiire 8: List of distributions to be simulated.

Probability paper

Choose probability paper: — Family 1 r— Problem

(i) Gumbel

O UJeibull

O Frechet

MaKima

O Minima

Rutomatic ticks

Title:

Cancel OK

Figure 9: Choosing probability papers.

Plotting point positions

Choose plotting point position method:

,— Method

(5) Mean

O Blom

O Hazen

O Gringorten

[ Cancel^

ID! Draujing 5 IHI

GUMBEL PROBABILITY PAPER FOR MAXIMA

0.9950

0.9900

0.9800

0.9500

0.9000

0.8000

0.7000

0.6000

0.5000 ^888

02000 m0.0050

0.00 0.20 0.40 0.60 0.80 1.00 X

Figure II: Sample drawn on probability paper.

Estimation

Choose one method:

I— Methods

(•) Probability least squares

O Return period least squares

O Standard least squares

O Maximum likelihood

O Percentiles

O Moments

lllhole sample

[ Cancel j [[ OK 'jj

Figure 12: Estimation methods dialog.

Figure 10: Choosing plotting point position formulas.

91
Distribution to be Fitted

Choose one distribution:

|— Family 1 i— Problem

(•) Gumbel

O lUeibulI

O Frechet

<•) MoHima

O Minima

( Cancel ) [[ OK |

Figure 13: Choosing limit model to be fitted.

Family : Gumbel

Plotting Point Method : Mean

Tail : right

Estimation Method : Maximum likelihood

First Order Statistic : 1 Last Order Statistic : 50

Location parameter : 0.353545

Scale parameter : 0.246368

Uariance-couariance matrix : 0.001356 -0.000331

-0.000331 0.000769

[ Delete ] [ OK

Figure 14: Estimates.

Domain of attraction

Choose one method:

r— Methods

(•) Pickands

O Curuature

Cancel

Finally, selection of the "domain of attraction"

option of the main menu leads to the dialog in figure 15 which allows us to choose the Pickands' or

the curvature method.

4.3 The expert system

The system can be used by inexperienced users to be guided in all the process of determining design

values, for example. The system starts by giving some information to the user who must answer

questions to the system. Depending on the answers,

the progress is conducted in different directions. As

one example, we have included the diagram in figure 16. Initially, the user is asked about whether

or not he is going to extrapolate available data. By

extrapolation we mean that the required values to predict the random variable are out of the range of

the observed values. This can occur either because

we are interested in large or small values of the

random variable or because we deal with very large or very small associated probabilities. If the user is not dealing with extrapolation, the problem is not an extreme value problem (limit) and standard

statistical methods can be used (see figure 16).

Next, we decide about dependence, independence or asymptotic independence. The latter is handled by means of the determination of a critical threshold

value above which independence can be assumed. In the first case the system is unable to solve the

problem and informs the user.

In the case of independence, we decide about which

is the tail of interest (left or right) and the system

draws the sample on the corresponding Gumbel

probability paper to make a decision about domains

of attraction. Depending on the pattern of the

drawing, the system recommends one of the classical

models or some alternatives.

A whole collection of decision trees, similar to the one above, can be easily incorporated to the system

to solve different extreme value problems. With the

guide of the system, the possibility of errors is greatly reduced.

4.4 Software implementation

Tne computer aided instruction package was

initially implemented on a Macintosh computer

using SuperCard. The statistical tools (simulation,

estimation, drawing and determination of domains

of attraction) were implemented as external

commands in Pascal language to allow for a direct access from SuperCard. Later, the convenience of a

Figure 15: Choosing one method to determine

domain of attraction.

92
Normal

Lognormal

Figure 16: Decision tree.

separate module was recognized and a different

program was developed using the Think Pascal

object oriented library. Communication between

both programs can be solved via Apple events.

• Simulation tools can be used to gain experience

and expertise on extreme value behaviour of

random variables.

5 Conclusions

From all the above we can conclude the following:

• Expert systems can be a good help to applied scientists and engineers to deal

with extreme value problems.

• Control of usual errors must be

implemented in expert systems to avoid

risks and/or waste of money.

• Recent developments of computer aided

instruction allows for an easy

implementation of teaching modules.

• Use of a set of decision trees leading to a guided determination of design values or

other extreme value problems can avoid

errors and leads to an important quality

improvement.

6 Acknowledgements

We thank the University of Cantabria and the

Direccion General de Investigacion Cientifica y Tecnica (DGICYT) (proyect PB9 1-0302), for partial

support of this work.

7 References

[1] Castillo, E. and Alvarez, E. Expert

systems. Uncertainty and learning. Computational

Mechanics Publications and Elsevier Applied

Science. London, New York. (1991). 331 pages.

[2] Weiss, S. M. and Kulikowski, C. A. A

practical guide to designing expert systems.

Kowman and Allanheld, Publishers. (1984).

93

Poisson Approximation Of Point Processes Of

Exceedances Under von Mises Conditions

Drees, H.

Universitiit zu Koln, Koln, Germany

Kaufmann, E.

Universitat-Gh Siegen, Siegen, Germany

Bounds on the Hellinger distance between certain truncated empirical processes and certain Poisson

processes are derived. These bounds depend, roughly speaking, on the rate at which a fairly general von Mises condition holds. Applying these results, also approximations of the joint distribution of

the k largest order statistics w.r.t. the variational distance are established.

1 Introduction

Let Xi, . .. ,Xn be i.i.d. random variables (r.v.'s) with

common distribution function (d.f.) F. Classical ex- treme value theory deals with the distributional the- ory and the asymptotic behaviour of sample maxima

Mn := maLX.{Xi, . . . ,Xn). From Gnedenko [7] one knows that, if the distribution of the standardized max- imum C{a~^{Mn — bn)) converges weakly to a nondegenerated distribution for some constants a„ > 0 and

bn G IR, then the limiting d.f. must be of the following

type (up to a scale and location parameter):

{exp(-2;-i/^) x>0, /? > 0, exp(-(-i)-i/^) if X < 0, /? < 0, exp(-e-^) 13 = 0.

Taking the logarithm and the first derivative, one obtains the functions

^0{x) := logG^(x) if Gpix) > 0,

r lx-i/^-H(o,oo)(x) /?>0,

M^) =

{ jk^-x)-''^-'k-ooM^) if /?<0,

I e-' (3 = 0

that will serve as mean value functions and intensity

functions of point processes. Moreover, if ^/3(x) > — 1,

1 -I- ^/3(x) defines a generalized Pareto d.f. with shape

parameter (3 and Lebesgue density il^p{x). The importance of generalized Pareto d.f.'s in extreme value the- ory was first pointed out by Pickands [15]. These d.f.'s play a central role in the present paper. More generally, one may deal with the k largest order

statistics, where k € {1, . .

. , n}, or with those observations that exceed a given threshold. In the second case,

one has to deal with point processes. For details about

point processes, we refer to Resnick [19] and Reiss [18]. The empirical point process based on the standardized r.v.'s a~^{Xi — bn), 1 < z < n, is defined by

n

i =l

where denotes the Dirac measure with mass 1 at x and 0 elsewhere. Given a threshold t £ IR, the empirical

point process of exceedances and a truncated Poisson

process are defined by

Nn,t:=Nn{-r)[t,oo)) (1.1)

and

at; := Ar*(.n[f,oo))

where N* denotes a Poisson point process.

In the following, we will prove results concerning the

strong convergence of distributions of truncated empirical processes, that is, convergence w.r.t. the Hellinger

distance. Recall that the Hellinger distance H and the

variational distance ||

• — • || between two probability measures Qo and Qi on a c-field C are defined by

and

IIQo-Qill := sup IQo(C) - Qi(C)| cec "1/ 1^0

95
where g,- is a /i-density of Q,-, f = 0, 1, and // is a measure

dominating Qo and Qi. Note that the variational and

Hellinger distances are topologically equivalent, yet the

rates of convergence in terms of these distances can be

of different order. The basic tool for dealing with the strong approximation of empirical processes is given by the following

two theorems.

Theorem 1.1 Let A^n.D := ^n(- H D) be a truncated

empirical process and N* jj be a Poisson process having

the same intensity measure as A^n,£>, D € B. Then

(i) \\C{N^,d) - C{N*,d)\\ < P{Xi e D},

(ii) H{C{Nn,D),C{N*^D)) < 3'/2p{Xi € D}.

For a proof of that result, see Theorem 1.4.2 in [18]. Theorem 1.2 Let , N2 be Poisson processes with

finite intensity measures j/* and 1/2, respectively. Then

(i) \\C{Nl)-C{Nm<^^l-^2\\,

For a proof we refer to [14], Proposition 1.12.1, or

[18], Theorem 3.2.2 and Theorem 3.2.1.

It was proved in [19] that weak convergence of the

sample maximum to an extreme value r.v. holds if, and

only if, weak convergence of the empirical processes Nn

to a certain Poisson process N* is valid. It was shown

in [6] that the corresponding result holds w.r.t. the vari- ational distance for the point processes of exceedances

Nn^t and the Poisson point process N* truncated left of

t > inf{x : G/j(ar) > 0}. Moreover, certain bounds for the accuracy of such approximations were established

in [17], [6] and [18]. The weak joint behaviour of several intermediate or- der statistics was studied in [1]. The strong asymptotic normality of single intermediate order statistics under von Mises conditions was proved in [4]. The ap- proximation of intermediate empirical point processes ^n{- n [<i, ^2]), truncated about the (1 — s/n)-quantile,

by a sequence of Poisson processes N*^*^^ , including the

homogeneous Poisson process, was investigated in [10]. There the accuracy of approximations was measured by

the rate at which a von Mises condition holds (see (2.1)-

(2.3)). This idea was also fruitfully utilized in [5] for

d.f.'s which are tail equivalent to a generalized Pareto

d.f.It was shown in [11] that multivariate maxima, defined by taking the maxima in each component, con- verge w.r.t. the variational distance if, and only if, certain truncated multivariate empirical processes converge. In that context, random thresholds are permitted. Approximation rates in the bivariate case were

established in [12].

The paper is organized as follows: In Section 2, the

density of F will be represented as the product of a

generalized Pareto density and a term depending only on a von Mises function. In Section 3, the truncated empirical process Ar„ < will be approximated by a Poisson process N*'" with mean

value function x —> s'ip{x), for x > t. That goal will be achieved in two steps by applying Theorem 1.1 and

1.2: first, we establish a rate of convergence of order

s/n for the Hellinger distance between Nn^t and the

Poisson process N* , having the same intensity measure

as Nn,f In a second step. A''* ^ will be approximated by

a 'limiting' process N*'' . Our main result in Section 3

will be the following:

Corollary 1.1 For t € Hi, let

( 1 + /3>0,

tp := { -(1 + /3t) if p<0,

[t f3 = 0.

Then, under the conditions of Theorem 3.1 and 3.2, we

have

HiCiNr.,,),C{N:p) = 0(i -h5^/2A„,^,,.,,)

for every fix t & IR with 0 < Gp{tp) < 1 uniformly for

|/?| <C<D-K

The term An^j , t^, that is defined in (3.2), measures

the rate at which a von Mises conditon holds. Moreover, the term s serves as a parameter of the stan- dardizing constants; for example, in case /? = 0, the

r.v.'s Xi, . .

. , Xn will be centered about the (1 — s/n)-

quantile. If the threshold t is fix, the expected number

of exceedances is proportional to s. In Section 4, we establish a bound on the variational

distance between the joint distribution of the k largest

order statistics cind a 'limiting distribution' if ib := [s] >

\ogn.

2 Representation of Density

Subsequently, assume that the underlying d.f. F possesses a density / on (it(F),u;(F)) for some u{F) <

uj{F) := sup{< : F{t) < l). Denote by a{F) := ini{t : F{t) > 0} the left endpoint of the d.f. F. It is well known (see, e.g. [17]) that F belongs to the strong domain of attraction of an extreme value distribution G/j

if one of the following von Mises conditions is satisfied

(for X T uj{F)):

(i) u{F) = 00 and

96
(ii) u>{F) < oo and

(2.2)

(iii) j;:///l-FHrf«< expand

f{x)f:^^>l-Fiu)du

^'^'•=

(1 - F(x))2 ^^^ = 0.

(2.3)

Moreover, put pp{x) := oo, z < u{F), and p^ix) := 0, z > w(F). The functions will be addressed as von

Mises terms. A necessary and sufficient condition for a d.f. F to belong to the weak domain of attraction of Go is the

existence of an auxiliary function U such that

l-F{xo + U{xo)x)

l-F{xo) exp(—x)

for every x and xo T <^iF) (cf. [8]). In this case, we may

choose

j:^''h-Fit)dt

1-F{x) U{x) := U{x) := (2.4)

for X < u{F) if J^^^^ 1 - Fit) dt < oo. Notice that

U{y) - U{x) = jl po(u) -ldu \{ u{F) <x,y< u;(F).

Moreover, define

Po,ro(a:) •=

po{xo + U{xo)x)

1 + Jo Po{xo + Uixo)t)- Idt

Uixo) = poixo + U{xo)x)

Uixo + U{xo)x)

if u(F) < xo,xo + L''(xo)x < u){F) and po,xo{x) := 0 elsewhere.

The density possesses the following representation.

Lemma 2.1 (i) Let 0 = 0, Xq e {a{F),u(F)), and

Xo + U{xo)x > u{F). Then

[/(xo)/(xo + U{xo)x) = rpo{x)pQ^xoix) exp ( - y po,ro(0 - 1 dt)

x(l-F(xo)). (2.5)

(ii) Let /? > 0, Xo > a{F), and xxq > u{F). Then

xofixox) = ^,(x)(/^p,(xox))exp(- j^^^^^MlLlZ^,,)

x(l-F(xo)). (2.6)

(iii) Let/3< 0, xq e (a(F),w(F)), and u>{F)-{u}{F)-

xo)x > u{F). Then

{u{F) - xo)/(u;(F) + {u{F) - xo)x) = V^(x)(-/?)p;,(c.(F) + (u;(F)-xo)x)

ppiuiF) + iu{F)-xo)t)-\-l/0

X exp J

x(l-F(xo)).

-t

df

(2.7)

Proof. We restrict our attention on (i), because (ii) and (iii) can be dealt with in an analogous way. We

have

1 - F(xo + C/(xo)x)

1 - Fixo) = exp

= exp

= exp

Fiv)

i-i:

po{xo + U{xo)v)U{xo)

dv

U{xo + U{xo)v) ) = exp(-x)exp ^ - y po^xoi^) - '^dv) (2.8)

if Xo, Xo + t/ (xo)x € {u{F),u{F)) and hence

U{xo)f{xo + U{xo)x)

, ^ Pojxo + U{xo)x) ,TTr^ \^\\ = ^(^°)

c/(xo + C/(xo)x) ^^-^^^° + ^^^°^^))

= exp(-x)po,ro(a;) exp ( - y po,xo(v) - 1 dvj

x(l-F(xo)).

Lemma 2.1 immediately yields an expansion for / in terms of a generalized Pareto density and a remainder

term h, that is, /(x) = xpp{x){l + h{x)) for sufficiently

large x. Moreover, conditions for tail equivalence may

be deduced. For example, in case of /? = 0, tail equivalence of the density / to an exponential density holds

if po,xo{x) -* a e (0, oo) and exp (- Po,io(0 -adt)

converges in (0, oo) if x ^ oo. In the following section, it turns out that the fac- torization in (2.5)-(2.7) in a generalized Pareto density

and a factor depending only on the von Mises term pp

is an useful tool for developing rates of convergence in extreme value theory.

3 Point Processes of Exceedances

In this section, rates of convergence are established for the Bellinger distance between distributions of point

97
processes of exceedances Nn,t and distributions of certeiin Poisson processes.

First let us point out the trade off between the ac- curacy of the approximation and the efficiency of statistical inference in the Poisson process model. If the

expected number of exceedances increases and the sample size is fix, then the information contained in the

limiting model increases, but the accuracy of approximation decreeises, and vice versa. For that reason it is of

interest to study the accuracy of approximation for several thresholds and standardizing constants such that

the expected number of exceedances increases when the

sample size n tends to infinity.

Given a d.f. F with density / on (u{F),u{F)), for some u{F) < u>{F), define

fO /?>0,

6„ := { u{F) if /? < 0, [F-^l-s/n) (3 = Q

and

F-^l-s/n) /?>0,

a„ := < u{F) - F-\\ - s/n) if /? < 0, U{F-'^{\- s/n)) fS = 0

where the function U is defined in (2.4) and s = s{n) €

(0,n). Hence the expected number of exceedances

E{Nn,t{IR)) = n(l - F{b„ + ant)) depends on 5 and

t. Applying Lemma 2.1, we see that the intensity measure of Nn,t possesses the Lebesgue density

nanfibn+Onx) = sV/9(a:)r/j,n(ic)x (3.1)

/?exp ( - /"(r^,n(«) - !//?)/« du), /? > 0,

\P\ exp ( - fl, (r^A^) + l/0)l{-u) du), p< 0, exp ( - ro,„(u) -Iduj, 0 = 0

for f>„ -I- a„x > u{F), where rp^x) := pp{bn + a„x),

13^0, and ro,„(i) := po.bA'')-

Denote by

An,/?,,,t := (3.2)

n min(t ,l),oo) \pp{x)-l/0l /?>0,

^"Pr6[6„+a„ min(l,- l),w(F)) \pp{x)+l/0\, f3<0,

S"Px6[&„+a„ min(t,0),w(F)) |Po(x) - 1|, 0=0

the distance of the von Mises term pp and its limit. Observe that An,p,3,t —» 0 if s/n —+ 0 and the von Mises

condition is satisfied for /?. In the following, the accu- racy of the approximations will be measured in terms of

An,/3,s,t- Notice that the term An,p,3,t is related to the

term A„ defined in [10], yet the normalizing constants

are different. In the sequel, the following notation is used for the

approximating processes. Let N* = N*'''^ denote the

Poisson process having the same intensity measure as Nn, and N*'' = N*'''^ the Poisson process with mean

value function x —*• s^p{x).

In Theorem 3.1, Nn^t is approximated by * , w.r.t. the Bellinger distance, where the error is, up to a constant factor, the expected number of exceedances di- vided by the sample size n. The conditions in The- orems 3.1 and 3.2 are introduced to keep the terms

Ci^,t, i = 1,2, defined in these theorems independent

of s, n, and F

.

To simphfy the notation, let (u;(F) — bn)/a„ := oo if u{F) = oo. Theorem 3.1 If 0 e IR, t e {a{Gp),u}{Gp)), D €

(0, 1), A„,^,,,, < D and, in addition, \t\ < 2-^A~{)(J,

if 0 = 0, then

HiCiNn,t),C{N:^t))<Ci,p,twhere

l-\IP+sign{t-\)D 0 >0,

(^_i^-i/P-sign(t+i)D if i3<:0^

e-'+5/^ 0 = 0. Proof. It follows from Theorem 1.1 that

H{CiNn,t), C(K^,)) < 3^/2 (1 _ Ff^bn + a„<)) •

First we are going to prove the case 0 = 0. Let x„ :=

2~^A~o''^,. Notice that in the case u;{F) < oo we have

bn + OnXn < u){F). This follows from

s:^^h-Fiu)du

1 - F{x) Uix) = < u{F) - X

as X T uj{F) and, thus, C/(6„) = - /j';;^^^^ Poix) -ldx<

(u;(F)-6„)A„,o,,,,. Hence {uiF)-bl)/a„ > >

x„. We have for x G [t,Xn) and some t?(x) G [—1,1]

|po,i„(x) - 1|

Poibn +anX)

1 + /o Po{bn + ar,u)-ldu

po{bn + a„x)

-

1

-

1

1 -I- t?(x)xA„,o,,,t < 2(l-H|x|)A„,o,,,t < 3D^/' (3.3)

and hence

I £Po,bM -ldu\< D^l^ -h 1/4 < 5/4. (3.4)

Together with s/n = 1 - F(6„) and (2.8) one obtains

1 - F(6„ a„<) = ^e-'exp^-^ PQ^b^{x)-ldx^

n

98
In the case /? > 0, use the identity 1 — F{an) = s/n

to show that for appropriate r? £ [— 1, 1]

1 - F(6„ +

exp

P/3{hn + anx)-l/0

dx

= Vi/^exp(-tfA„.^,,,,^*irfx) _ £^-l/^-^A„,;9.,,t n <

if-i/^+sign(t-i)i? ~ n The proof of the case ^ < 0 can be carried out by

similar arguments.

In a second step, the Poisson process A'^^ , will be

replaced by the Poisson process A7'''- Theorem 3.2 If P e IR, t e (a(G/j), ^(G/j)), D e

(0, 1), An,0,s,t < D, > \p\ and, in addition, \t\ <

^~'^:!o,lv if 13 = 0, then

where

C2,0,t := mi-^0it)yf^ + ii + \0\Dy/^x (3.5)

\V2

xmaxi\x\^,\x\-^)dxj ,

[f.'^iil + lxlfe-^dx + iSM^y'

+ ( + \x\)h^/\l + ZD'/^)e-- dx

+(8/e)4e5/^J , 0 = 0.

Proof. Let

L0{x) := f'{p0{hn-¥ar,u)-\/0)/udu, p>0,

Jl,{p0{br, + ar^u) + l/P)/{-u)du, 13 <0,

Jo Po,bAu)-ldu, (3 = 0

if< <x< (u;(F)-6„)/a„.

First we prove the caise 0 = 0. Using Theorem 1.2, one obtains that the Bellinger distance between Pois- son processes is bounded by the Hellinger distance of

the corresponding intensity measures. Let iV' be a Pois- son process whose intensity measure has the density X -* spo,i„(z)e"''l[t,x,)(x) with Xn := 2-^A„ o.j.t- Applying the triangle inequality and (3.1), one obtains

< (y"~ ((na„/(6„ + a„x))i/2

, \ 2 \ 1/2

-{spo,Kix)e-'Mt,r.)i=^))'^') dx)

+ (/~ ((^^o,6„(x)e-n[,,.„)(x))i/2

= J

^exp(-Lo(x)) - l) po,i„(x)e"''dx

N, 1/2 +n(l - F(6„ -Ha„x„))j

/ \ 1/2 +

J

ipo,iM'^^-'^fe-'dx + se---)

say.

Taking into account that (x^/^ — 1)^ < (x — 1)^ and < {ex/A)~'^ , X > 0, we obtain from (3.3)

h < j'\po,iSx)-lfe-^dx + e-^-

= _^""(2(1 + |x|)A„,o,,,t)'e-' dx + (ex„/4)-^

< A^,o,,,, 4(1 + \x\fe-' dx + (8/e)^) (3.6)

Combining (2.8) and (3.4) leads to

n(l - F{hn \- anXn)) — se""^" exp ^ — y po,i„(«) — 1 rf")

< 5A2,o,,,,(8/e)^eV^ (3.7)

A Taylor expansion yields

/i-(n/s)(l-F(6„-ha„x„)) = y

" (exp(-Lo(x)/2) - l)'po,6„(x)e-" dx

= (-Io(x)/2)'(exp(-t?(x)Lo(x)/2))'

xpo,6„(x)e~^ dx < Alo,s,t /'"(I + \x\)'e'/\l + ZD'f')e-^ dx

(3.8)

for some d{x) € [0, 1] where the last inequality follows

from (3.3) and (3.4). Combining (3.6), (3.7), and (3.8), we get i/(£(Ar* ,),£(A^;'0) < C2,o,«s'/'A„,o.,,t with

C2,o,t defined in (3.5). Next, we turn to the case (3^0. Check that

|I^(x)|< A„,^,,,,|log|x|| (3.9)

holds for /? 7^ 0 if t < X < w(G^).

99
Let N he a. Poisson process whose intensity measure has the density x —> sp^{hn + anx)|/i3|

x^p{x)l]^t,u>{Gfi)){x). Arguing as in the case /? = 0, one obtains

H{C{K^,)X{Nr))

< H{C{K^,), C{N)) + H{C{Nl C{N:''))

((na„/(6„+anx))i/2

+

( f^""'^

((sp^(6„ + a„x)|/?|V'/j(x))^/2

-(5V'^(x))^/2) dx)

, r<^(Gp) V 1/2 X 2 = [s (exp(-L^(x)) -l)

\ 1/2 xp/3(6„ + a„x)|/?|i/'/j(x) dxj

+ (5^''^'''\(|/?|p;5(6„ + a„x))i/2-l)2

\ 1/2 xipp{x) dxj = .^/27;/; + .^/^4/;, say.

Moreover,

{\0\pp{bn + ar,x)-l)^lPp{x)dx

Applying a Taylor expansion and (3.9), we obtain

= ^"^'''^(exp(-Mx)/2)-l)'

xpp{bn + a„x)|/?|V'/?(x) dx = {-h{x)/2)

x(exp(-i?(x)I^(x)/2))'

xp^(6„ + a„x)|/?|t/'/3(x) dx

(-(log|x|)/2)

X exp ^A„,/j,,,t|log|x||)

x{-^^+D')\P\Mx)dx

[

' (^^)'^/5(^)niax(|x|^,|xl-^)cfx

for some i?(x) G [0, 1].

The rates established in Theorem 3.1 and 3.2 are sharp if s > 51ogn. This follows from results in Section 4, where it is shown that for the k largest order

statistics (ib = [s/5]) the same rates of approximation

hold as for the point processes and that the rates obtained for the k largest order statistics are sharp. In the

case /? = 0, similar bounds to that in the Theorems 3.1 and 3.2 are established in [2] for different normalizing

constants.

Proof of Corollary 1.1. We have to show that

there exists a constant C > 0 such that Ci,^,t < C, z =

1,2. But this is immediate from the well-known formula

(1+^x)-^/^ e-' as /? ^ 0 and some straightforward

calculations.

Using Theorems 3.1 and 3.2 (respectively Corollary

1.1), one may establish rates of convergence for point

processes of exceedances if the d.f. F fulfills one of the von Mises conditions (2.1)-(2.3). The proofs of the fol- lowing examples may be carried out by elementary cal- culations (cf. [10]). Example 3.1 If F{x) := 1 -f- 'J'^(x), *^(x) > -1, de- notes a generalized Pareto d.f. for some /? G iR, then

i/(>C(Ar„,t),£(iv;''))<3^/2(-*^(0)^

for f € (a(G^),w(G/3)).

Since for the generalized Pareto distribution the ap- proximating Poisson process in Theorem 3.1 equals the process N'^'' , the bound in Example 3.1 is, up to a constant factor, equal to the expected number of ex- ceedances ENt{IR) divided by n. The following example was first proved in [6], Theo- rem 4. Example 3.2 (6-condition)

Assume that for some f3 E IR, S > 0, and L > 0 the

density f has the form

f{x)=^pix)e'^^-'> (3.10)

where \h{x)\ < L(-*^(x))^ Then

H{C{Nn,),C{Nr)) = 0{^ + s'/'{^y) (3.11)

for every fix f € (a(G^),u;(G/j)).

Example 3.3 (Normed distribution) Let F be the normal d.f. Then

for every fix < € 2R.

100
4 Joint Distribution of Order

Statistics

In this section, we study an application of the preceding results on point process approximations to derive

rates of convergence for the joint distribution of the ib largest order statistics. Expansions of the d.f. of sample maixima under von Mises conditions may be found

in Radtke [16]. Uniform convergence of order statistics under von Mises conditions was studied by Falk [3]. Sweeting [20] has shown that the von Mises conditions

(2.1)-(2.3) are equivalent to the uniform convergence

of densities of maxima on finite intervails. For a comprehensive treatment of order statistics and, in particular, approximations of intermediate and extreme order

statistics w.r.t. the Bellinger distance, we refer to Reiss

[17]. There a method was introduced for establishing

rates of convergence of point processes from rates for the ib largest order statistics. We use that method in the converse direction. In the following, we define for a point measure ^ on

Ft and k G {1, . .

. , n} the terms

mfc,t(^) := inf{x > t : oo)) < k]

and

Mk,i{fi) := (mi,,(/i), . .

. , mfc,t(/i))

Notice that, for the order statistics Xi-n < < Xn-n

ofXi ,

. . . , Xn and for the empirical process Nn^t defined

in (1.1), the identity

(a-'{Xn-i+l.n - 6n))'^^ = MkANn,t)

holds if a-^{Xn-k+l:n - K) > t. To obtain sharp bounds from Theorems 3.1 and 3.2, one has to find conditions such that P{a~^{Xn-k+i:n —

bn) < t} is of the same order as the error of the point process approximation. For that purpose, we use an

exponential bound for single order statistics, which is a consequence of [17], Lemma 3. 1.1. For a proof of

Lemma 4.1 see [9]. Lemma 4.1 Lei Ui,...,Un be independent and uniformly disiribuied on (0, 1), i € {1, • • • , n} and C > I. Then

P{Un-,+v.n < 1 - Ck/n] < exp ( - k^^=-^)

.

Notice that in Lemma 4.1 a bound of order 0{n~^)

may be achieved by choosing ib > log n and C > 5. Denote by Qt,^ := C{Mk-co{N''')) the joint distri- bution of the ib largest order statistics of the Poisson

process N'-' with mean value function x —> s*/3(x), X > a{Gfi). Our main result in this section is the fol- lowing:

Theorem 4.1 Lei k ^ {1,..., n}, k > logn, C > 1, s = Ck, andip := sign{f3). Then, under the conditions

of Theorem 3.1 and 3.2, we have

|l£(a-^(X„_Hl:n-tn))Ll-Qfc,/3ll

< 2f„-^i^-h3i/V/4^ ~ \ n

¥C2,0,uC''''k'l''/^r.,0,ck,t,) (4.1)

where C2,p,t0 is defined in (3.5).

Proof. Recall that F possesses a density on

{u{F),u{F)). The condition A„,^_cJt,t^ < D implies

u{F) < a^tp + hn = F~'^{\ - s/n). Hence F is con- tinuous at F~^(l — s/n). Denote by Un-k+v.n the kih. largest order statistic of n independent r.v.'s which are uniformly distributed on (0, 1). Since £(Xn-jt+i:n) =

£(F-^(i7n_fc+i:n)), We may write

P{a-\Xr,-k+V.n-bn) < tp) = P{Xr^-k+l:n<F-\l-s/n)} = P{F-\Un-k+l:n) < F-\l- s/n)} = P{Un.k+V.n <l-s/n}.

Using Lemma 4.1, we get

P{a-\Xr,^k+l:r^-br.)<tp}<n 3C(n +l)^ (4.2)

Recall that the Bellinger distance dominates the vari- ational distance. The triangle inequality and the monotonicity theorem (see, e.g. [13] or [18], Lemma 1.4.2)

yield

||£(a;^(X„_,+i:„-6„))Li-QMll

< ||£(a-i(X„_.+::„ - 6„))Li - £(Mfc.t,(iV„,,J)||

+mMk,t,{Nr.,t,)) - C{Mk,t,{N:p)\\

+\\C{Mk,t,{N:;))-QkM\

< P{a-\X„.k+l:n-bn)<tp}

+\\C{Nn,t,) - CiKpW + P{N,y{IR) < k).

Since P{N';'iIR) < k) < P{Nn,ti,{IR) < k] +

ll^(^n,f^) - C{N'p\\, the assertion follows from (4.2)

and Theorem 3.1 and 3.2.

Notice that mjt,_oo(A'^*'*) possesses the d.f.

P{mk,-oo{N*'') < x} = P{N'''({x,oo)) < ib - 1}

- Gpi^)Z^ J, .=0

for X >a(G^) (cf. [18], E.6.2).

101
If ^ > logn, the rate obtained in (4.1) is sharp for distributions treated in Examples 3.1 and 3.2 with 6 =

1. That follows from [17], Theorem 5.4.4 and Example

5.5.6. Sharp rates for every k € {l,...,n} may be

derived by direct calculations using Theorem 5.5.4 in

[17].In the case of normal r.v.'s Theorem 4.1 yields more

accurate bounds than those known in literature if A: >

logn (cf. [3], Example 4.53).

Example 4.1 Lei Xi,...,Xn be independent, nor- mally disiribuied r.v. 's and k > logn. Then

||£(a-i(X„_,+i:„ - 6„))Li - Qk.oW = 0{^^)

where On and bn are chosen as in Section 3 with s := hk.

Proof. The proof follows from Theorem 4.1 and Example 3.3.

References

[1] Cooil, B., Limiting multivariate distributions of

intermediate order statistics. Ann. Probab., 13 (1984), 469-477.

[2] Drees, H., Refined estimation of the extreme value

index. Ph.D. thesis. University of Siegen, 1993.

[3] Falk, M., Uniform convergence of extreme order

statistics. Habilitatonsschrift, University of Siegen,

1985.

[4] Falk, M., A note on uniform asymptotic normality

of intermediate order statistics. Ann. Inst. Statist. Math., 41 (1990), 19-29.

[5] Falk, M. and Marohn, F., Von Mises conditions

revisited. Ann. Probab., 29 (1993), 1310-1328.

[6] Falk, M. and Reiss, R.-D., Poisson approximation

of empirical processes. Statist. Probab. Letters, 14

(1992). 39-48.

[7] Gnedenko, B., Sur la distribution limite du terme

maximum d'une serie aleatoire. Ann. Math., 44

(1943), 423-453.

[8] Haan, L. de. On Regular Variation and its Application to the Weak Convergence of Sample Extremes. Math. Centre Tracts 32, Amsterdam, 1975.

[9] Kaufmann, E., Contributions to Approximations

in Extreme Value Theory. Ph.D. thesis. University

of Siegen, 1992.

10] Kaufmamn, E. and Reiss, R.-D., Poisson approximation of intermediate empirical processes. J. Appl. Prob., 29 (1992), 825-837.

11] Kaufmcinn, E. and Reiss, R.-D., Strong con- vergence of multivariate point processes of ex- ceedances. Ann. Inst. Statist. Math., 45 (1993),

433-444.

12] Kaufmann, E. and Reiss, R.-D., Approximation

rates for exceedances processes. To appear in: J. Statist. Plann. Inference (1994).

13] Liese, F. and Vajda, I., Convex Statistical Distances. Teubner-Texte zur Mathematik, Bd. 95. Teubner, Leipzig, 1987.

14] Matthes, K., Kerstan, J. and Mecke, J., Infinitely

Divisible Point Processes. Wiley, Chichester, 1978.

15] Pickands, J. Ill, Statistical inference using extreme

order statistics. Ann. Statist., 3 (1975), 119-131.

16] Radtke, M., Konvergenzraten und Entwicklungen

unier von Mises Bedingungen in der Extremwerttheorie. Ph.D. thesis. University of Siegen, 1988.

17] Reiss, R.-D., Approximate Distributions of Order

Statistics: With Applications to Nonparametric

Statistics. Springer, New York, 1989.

18] Reiss, R.-D., A Course on Point Processes.

Springer, New York, 1993.

19] Resnick, S.I., Extreme Values, Regular Variation,

and Point Processes. Springer, New York, 1987.

20] Sweeting, T.J., On domains of uniform local at- traction in extreme value theory. Ann. Probab., 13

(1985), 196-205.

102
Estimating The Extremal Index Under A Local

Dependence Condition By The Reciprocal Of

The Average Length Of Successive Runs

Duarte, L.C.C.

University of Lisbon, Lisbon, Portugal

Abstract: Whenever a strictly stationary, strongly mixing sequence, satisfies the local

dependence condition D"(Un) of Ref. [1], the point process defined by the upcrossings of a high level Un has an important contribution to the characterization of the limiting compound

Poisson process of exceedances.

For each n let N(n,Un) =2^i ll(Xi>Un) be the random variable that represents the number of

exceedances of u^ in a sample of size n, and N(n,Un )=Xj^]^ 11 (Xi.i<Un<Xi) the number of

upcrossings of the same level. Then, if we consider levels u^^ix) such that nP[Xi>Uii(T)]->T,

when n-»oo, the extremal index 6 verifies

nP[Xi>Un(T)] E[N(n,Un(t))] 1 lim —p T=lim—pi T =Q • n-^ nPLXQ<Un(T)<XiJ n^oo E LN(n,Un(T))J ^ Now, if the sequence of levels is such that E[N(n,Un)]—>l,when n—><», then the reciprocal of the

extremal index will merely be the limit of the mean number of exceedances. Based on this

result we have developed a method of estimation of 6 that consists on dividing the sample in kn

blocks of size r^ and taking the average number of exceedances of the level Uni, suitably

defined for the i-th block, so that the mean number of upcrossings of this level in that block is approximately 1. More precisely, we present here some properties of the estimator

^ IitlNi(rn,Uni)Vl

9n =

1 . Introduction

In this paper we study an estimator for the extremal

index which has been motivated by Ref. [1] and for this reason we will assume for the underlying

stationary process, conditions analogous to those

used by these authors. In order to make clear the origin of this estimator we

start with the presentation of some known results,

related to this subject.

Given a stationary sequence {Xj}, i>l, and denoting

by y|(u) the a -field generated by the events

{(Xj^<u):i< k <j}, the following mixing coefficients

will be used

^n^-^'^) = sup { I P(AnB) - P(A)P(B) | :

These are related to the well known long range

dependence condition A(un). More precisely, {Xj} is said to satisfy A(Un) if, for some i n=o(ii)>

(Xn(/Jn>Uii)"*0 as n->oo. This condition is stronger

than D(Un) defined in Ref. [2], but it wiU be needed to validate the convergence of some point processes

important in this work.

103
Concerning the local dependence structure of {Xi} we

assume henceforth the validity of the condition D"

defined in Ref. [1] , which, in some way, restricts the

local occurrence of two or more upcrossings of high

levels.

Let kn be a sequence of integers, with kn— and

such that knan(in.Un) 0, kninii^O and

knP(Xi>Uii) -4 0 . We say that (Xi) verifies D"(un) if

for ^n=|^^j we have

lim n S P[Xi>Un,Xj<Un<X.^i] = 0. n^oc j=2

Note that we say that {Xi } has an upcrossing of the

level u at j if Xj.i<u<Xj . So, if we represent by |J,(u) the

probability of such an event (which is independent of

j by stationarity) then |x(u) can be interpreted as the

mean number of upcrossings of u per unit time.

Considering that for a stationary sequence

^i(u) = P(X^<u<X2) = P(X2<u I Xi>u) P(Xj>u)

we can conclude that for sequences with the same

marginal distribution, the lesser the value of |J. for a fixed level u, the stronger the tail dependence

structure near that level. Notice that the extremal index of {Xj } is equal to 9 if for each sequence of levels Un(x) such that

nP[Xi>Un(i:)]-»T the limit of P[Mn < Un(x)] is e-^'^,

(here Mq denotes the random variable max^jfXi}).

This parameter 6 takes values in the interval [0,1]

and measures the strength of dependence of a stationary sequence. The stronger the dependence

the lesser the value of 6 so that for an i.i.d. sequence

we have 9=1 whereas the value 9=0 corresponds to a long memory sequence. In our study we admit a weak dependent structure for the sequence in such a way that its extremal index, when exists, is strictly

positive.

The connection between the last two paragraphs is quite clear and it can be proved that for a suitably

chosen sequence of levels Un, for which both

conditions A and D" hold, the extremal index can be

obtained as the limit of P(X2^Un | Xi>Un) when n

goes to infinity. This result comes straightforward

from the following proposition established in Ref.

[1]:

Proposition 1. Suppose that A(Un) and D"(Un ) hold for some sequences {unl, (kn) and (rn) satisfying the

conditions mentioned above. Then

P[Mn ^ Uq] -» e"^ if and only if n|x(Un) v . Hence, when the process has an external index 9, we

have

nn(un)-*v if and only if n P[Xi>Un] v/9 (1)

In the sequel we wiU also be interested in two point

processes relevant to the development of an

estimator of the extremal index. The first one is the

(time normalized) point process N^ of exceedances

of a high level Uq:n Nn(B)=Ie^^(B)ll(Xi>u^), Bc[0,l].

This point process has been fuUy studied in Ref. [3], and it is shown, for example, that under A(Un ) any

existing limit of N^ must be a compound Poisson

point process. When Un=Un(x) is such that

nP[Xi>UrL(t)] -» t, it is proved, under general

conditions, that the underlying Poisson point

process has intensity 9t and the mean value of the

multiplicities is 9"-'-. From the proof we can infere

that the Poisson points can be regarded as positions

of clusters of exceedances and the number of

exceedances in each cluster corresponds to the

multiplicities.

Let 7tn(j) be the distribution of the number of

exceedances in a cluster given that there is at least one exceedance

f'^ . 1 7rn(j) = P|5^fl(Xi>%) = j I 5^11(Xi>u^)>0|,

i=l i=l

j=l, 2, ... The result mentioned above can be more precisely

stated as follows:

Proposition 2. Assume that A(Un) holds for the

stationary sequence {Xi} and that, for some v>0,

limP[Mn<Un] = e'^. n-x» Suppose there exists a probability distribution 7c(j) such that Tz(j) = lim7tn(j), j=l, 2, where n^(j) is

n-><» the conditional probability distribution of the

number of exceedances of Un defined for blocks of

size rn=[n/kn], with kn going to infinity in the

conditions stated above. Then N^ converges in

distribution to a compound Poisson process with

intensity v and multiplicity distribution 7i(.).

As we have seen, for a stationary sequence with

extremal index 9 it is possible to consider

normalized levels Un(t) in such a way that we have

At

lim P[Mn< Uq] = e' . For this kind of levels the

104
mean number of exceedances in a sample of size n

is assimptotically t, since

n. E{Nn([0,l])} = I ^y^i[0,l]) E{ll(Xi>Un(T))}

i=l = nPPCi>Un(T)],

and in turn, if A(un(t)) holds, it is proved in Ref. [4] that the mean cluster size is approximately Q'h

lim 7i:n(j)= Q'^-

^"*°°j>l

Now, the size of a cluster induced by Un(x) is just the

number of exceedances of that level in a block with

size rn, conveniently chosen. A natural estimator of

e'-'- would then be obtained by dividing the total

number of exceedances of a previously defined high

level by the total number of clusters. This procedure

is quite general and has been proposed in Ref. [5] . It has the (not so minor) problems of choosing a convenient block size and a convenient high level. As we will see, the first problem can be avoided if we

assume a local dependence condition like D". This

is due to the important role played by the point

process of upcrossings in the characterization of the

limiting compound Poisson process of exceedances.

According to Ref. [1] the point process of upcrossings n Nn(B)=I (B) nCX. 1 <Un<Xi)), BC[0,1] , i=l converges to a Poisson process whose intensity

depends on the chosen sequence of normalized

levels and on the value of the extremal index.

Projiosition 3: Suppose A(Uii) and D"(Un) hold for a sequence of levels u^ such that the mean number of

uppcrossings is approximately v, i.e., n(i.(Un)-»v.

Then N^-^N, where N is a Poisson process in [0,1], with intensity v.

If, in addition, {Xi } has an extremal index 9, (1)

holds and, consequently, 6t is the intensity of the

limiting Poisson process N, generated by the levels

Mj^ix) such that nP[Xi>Un(T)]-»T.

Under the conditions mentioned above the

equivalence (1) gives us two ways of constructing

normalized levels. In the following we denote by

Un(5) levels such that n|x(Un(6))^ 8, and by Un(8)

levels such that nP[X-,^>Un(8)] 5 vso, in case of

existence of the extremal index, Un(8) =Un(e8)).

Another consequence of assuming D" is that it enables us to identify clusters of exceedances with

runs of consecutive exceedances.

Given a sequence of levels Uq , define for each n the

random variables Yj=Y(j,Un), l<j<n, which

represent the number of consecutive exceedances of

the level u after time j, {Yj=0} = {Xj<Un}

{Yj =k}={Xj >Un >Un ,-,Xi+k. i >^n ^+^^-^n > 1^

Denote by Zj (Uq ) the length of a run of consecutive

exceedzmces after the occurrence of an upcrossing at time j, and represent by

Sn(k) =P[Zj(Un)=k]

= P[Yj=k/Xj.i<Un<Xj],k>l,

its probability distribution (which does not depence on j, due to stationarity).

The following statements established in Ref. [4] summarize a few results that are essential to our work.

Proposition 4. Let {Xj } be a stationary sequence with

extremal index 6. If Uii=Uii(x) is such that D(un) and

D"(un ) hold then

limE[Zi(Un)] =e"^

Proof: In order to simplify the notation, let Z denote the

random variable ZjCun). Since Z takes only positive

integer values,

E(Z) = I (l-F^Ck-D) = S P[Z>k]. (2)

k>l k>l

But Z represents the length of a successive nm, so

P[Z>k] = P[Xi>Un;X2>Un;...;Xj^>Un iXo<Un<Xj

P[Xq<Uii; Xj>Un; Xj^>Un ] P[XQ<Un<Xj

P[X^>Un;...;Xj^>Un] - P[XQ>Un;...;Xj^>Un]

P[Xo<Un<Xj

P[Yi>k] - P[Yi>k+l]

P[Xo<Un<Xi]

Hence the mean value of Z can be easily calculated

fi-om (2)

105
I(p[Y,>_k]-p[y,^k.i])

p[ J E(Z) = F = — •

The result follows immediately since P[Yj>i] =

P[Xi>Un] and Un=Un(T) (=Un(eT)).

Proposition 5. If Un is a sequence of normalized

levels for which both conditions DCu^) and D"(uii) are verified then, for each k=l,2,...,

lim[7Cn(k) -TCn(k)] = 0.

After this result tlie following proposition, analogous

to proposition 2, becomes quite apparent:

Proposition 6. Assume that A(un ) and D"(un) hold

for a sequence of levels Uii=Un(v), with v>0. Then, if Zi(un) converges in distribution to some non

degenerated random variable Z, the point process of

exceedances converges vaguely to a point process

N such that

N(B)

N(B)= ^^Zj, Bc[0,l],{Z-}i.i.d. with Z,

i=l where N is the existing Poisson limit of Nn . Hence

the point process N is a compound Poisson process

with intensity v and multiplicity Z.

The estimation of the extremal index presented in

Ref. [4] is based in these last results. From

proposition 4 we see that in the limit the cluster

centers can be identified with the upcrossings

whereas, by proposition 1, the mean size of each

cluster is aproximately Q'^. The suggested estimator

for 6'-'^ is then constructed by dividing the total

number of exceedances of a conveniently chosen

level by the total number of upcrossings of that level.

2. An estiinator for the extremal index,

under D"

Given a sequence of levels u^ define for each n the

random variable N(n,Un ) which represents the

number of exceedances of Uq in a sample of size n,

i.e..

n N(n,Un)= ^ll(Xi>Un)i

1=1 Note that N(n,Un ) is nothing but the measure of the

interval [0,1] through the time normalized point

process Nq.

In a similar way we denote by N(n,Un ) the number of

upcrossings of u^ in a sample of size n, i.e., n N(n,Un)=X KX. .<n^<X.).

i=l Now, the mean values of these variables are E(N(n,Un))=nP(Xi>Un) and E(N(n,Un ))=nn(Un),

respectively. So, under D"(un) and if the extremal

index exists, we have, by (1),

lini

E[N(n,Un)] ^1

n-S. E[N(n,Un)]

for any sequence of normalized levels. If we

consider the sequence of levels in such a way that

E[N(n,Un)]-^l,when n—>o°, then the reciprocal of the

extremal index wUl merely be the limit of the mean

number of exceedances.

For i=l,2, kn, let Unj be a sequence such that

rn|x(Uj^-)-»l (recall that k^ is the number of blocks of

size rn in which we subdivide the sample). Represent

by N-(rj^,Uj^-) the number of exceedances of Uj^- in

block i, and consider the following estimator for the

extremal index

'en=(^l|SlNi(rn,u^)^

To prove the consistency of On we use the assimptotic

independence of anNi(rn,Uii), i=l,2, k^, for any

sequence of real numbers {a^}, under the validity of

the condition

kn[cxn('«n-2,un) + P(M, >Un)] -* 0 (3)

for some in (Lema 5.2.1, Ref. [4]).

Proposition 7. Let kn be a sequence of integers such

thatkn—>«> and let rn=[n/kii]. Suppose that the

stationary sequence (Xj} has an extremal index 9 and that for a sequence of levels Un verifying

rn^(Un)-»l, A(Un), D"(Un) and (3) hold. Then

kn

X Nj(rn,Un)

1=1 R 1 kn e

imder the condition that

106
lim E[N2(rn,Un)]=c2 <oo. n—

Proof:

Consider a sequence {N • (rn,Uii)}, i>l, of

independent random variables identically

distributed with Ni(rn,Un)- Taking an=l/kn in

Lema 5.2.1, Ref. [4], we have the assimptotic

independence of Ni(rn,Uii)/kn and the following

convergenge

lNi(rn,Un) I Nf(rn,Un)

i=l 1=1 4o.

kn kii On the other hand, since for this kind of levels we

have lim E[Ni(rn,Un)] =9"'^, the result follows n

immediately from the convergence in probability of

X Nf(rn,Un) /kn to 6"^. Indeed

i=l

P [ I ir I Nt(rn>Un) - E [N^Crn.Un)] | >e]

i=l

= Ptr I l(N*(rn,Un)-E LN*(rn,Un)]) | >e]

Tin i=l

var [ X N*(rn,Un)]

i=l = o(^)

by the asymptotic boundness of var [N*(rn,Un)].*

Proposition 8. If conditions of proposition 7 hold for

levels Uni, such thatrn^.(Uiii)->l, i=l,2,..., k^, then Gn

is consistent.

Proof:

Let Un and 1% be two sequences of levels such that

rn^(Un)-»l and rnjidt)-*!- Then, using (1) we derive

P[N(rn,Un)^N(rn,vt)] <rn

I P[Xi<Un] - P[Xi<uJj

| = rn (l+od)) - ^

(1+0(1))) = 0(1)

err when n goes to infinity.

Consequentely N(rn,Un) - N(rn,it) 0 and the

consistency of 0n follows from the identity

1%

X N-(rn,Un)

i=l

X (N-(rn,Un)-N-(rn,Uni)) ^ — z +(en)

where the right hand side converges in probability to

1/6, by proposition 7, and the first term in the left hand side goes to zero, in probability, by the above

remark.

The Lindberg condition for the asymptotic

normality of (Xq = l/9n is easily established from

the fact that is the arithmetic mean of

approximately independent random variables.

Proposition 9. Under the conditions of proposition 7

if, for each e>0,

lim kn E [NVn.Un) ll(N(rn,Un)>£)] = 0

n-»«

( Lindberg condition)

then

, -1/2 kn

r hi > XNi(rn,Un)-knE(N(rn,Un))

i=l

JS) N(0;|Ve2c2-l) (4)

Proof:

Let {NJ(rn.'iin)}) i^l. be a sequence of i.i.d. random

variables with the distribution of N(rn,Un). This

sequence verifies the conditions of Theorem 3 of

Ref. [6], p. 101, from which, using the convergence of

E(N(rii,Uii)) to 1/e, we conclude that

-172 / kn

I (Nf(rn,un) - E(I^(rn,Un)))

i=l

J9

The asymptotic distribution (4) follows immediately

since

107
lNi(rn,Un)-knE(N(rn,Un))

i=l J

N(n,Un)-INf(rn,Un)

i=l

l(N*(rn,Un)-E(Nf(rn,Un)))

^i=l and by Lema 5.2.1, from Ref. [4], -1/2

N(ii,un)- IN7(rn,Un) converges in

distribution to zero.

Remark: Both the consistency and the asymptotic

normality of this estimador strongly depend on the

applicability of Lema 5.2.1 from Ref. [4], in which

condition (3) is assumed. It is easy to see that, for normalized levels Un, A(un) is sufficient to (3), but

for practical proposes it is worth while to remark

further that, if rn|x(Un)-» 1, then

knP(M, >Un)>kn PCX, >Un)^ (9 + 0(1)).

So, in this case, k^/rii-^O is a necessary condtion to

(3), that is, the number of blocks must be

significantly smaller than the size of each block.

Proposition 10. Suppose in addition to the conditions

of proposition 9 that

A£~(Ni(rn,Un)-Ni(rn,Uni)) Ao. (5)

for levels Uni, such that rn^(Uni)-*l, i=l, 2,..., kn.

Then

VkTC en - en)

^ N(O; eV e2c2-l), (6) where en= (E[N(rn,Un)])'"^.

If in addition en= 6 + o(—==), then the following

convergence also occurs Vk7( en - e)

^

N(o; eVe2c2-l). (7)

Proof:

Let ttn = 1/en and ttn = E(N(rn,Un))- After some

rearrangements we have

/k^(o(n - On)

-1/2 = kn S Nj(rn,Un) - knttn

i=l

-k

1/2

X (N-(rn,Un)-Ni(rn,Uni))

fi-om which, using (4) and (5) we first conclude that

e The asjrmptotic distribution established in (6) can

easily be obtained using for instance the 5-method

(Ref. [7]). The convergence in (7) is trivial after the

decomposition of the first member of (6) into the

terms A/k^( en -e) and V^J^ e-Qn)-*

3. Simulation results

In this section the results of a simulation study are presented, in which we choose for {Xi} the maxautorregressive process studied in Ref. Ref. [8]. More precisely, let Xi=cmax(Xi.i,Yi), cg(0,1), where

the r.v.'s Yj are i.i.d. with a Frechet distribution

function with parameter a. This process has a stationary distribution, verifies conditions D and

D" and has an extremal index e=l-c°'. Further, it can be proved that the limiting cluster distribution is Geometric with parameter e and so, if we denote this

r.v. by Z, (in accordance with the notation of section

1), we have E(Z)=iye, var(Z) =(1-6)62.

Now, in most practical cases, the levels Un

satisfying the condition rn P[Xi<Un<X2]~l are

typically unknown since they depend on the

knowlege of the joint distribution of (Xi,X2). As in

other methods we will consider the replacement of

these deterministic levels by random ones in

accordance with the above relation. So we suggest the

following strategy: — Choose kn (the number of blocks) in such a way

that [n/kn]=rn>kn. — In each block pick up all the maximum terms

within the points j such that Xj.i>Xj , Xj+i >Xj

.

— For block i, consider Uni the second greatest

value among these maximum terms.

In this way there will be in each block at most one upcrossing of this level which will be as low as

possible. In most cases the level Uni corresponds to the second highest peak in block i, as illustrated in

fig.l.

As n goes to infinity it is reasonable to think that the

exceedances of Unj occur in a cluster that is asymptoticaly independent of the cluster that

contains Uni- Thus, the next result supports our

choice of Uni108
Figure 1: Construction of the random levels Unj

Proposition 11. Let {X^ be a stationary sequence with

extremal index 6 and a marginal continuous

distribution F. Let =niax(Xi,X2,...,Xji ) and

consider a r.v. X with the same distribution F,

independent of {Xi). Then lim nP(X>Mn) = 6'^. n—»oo

Proof:

Denote by the r.v. nF(Mn), where F=l-F. By the

definition of the extremal index it follows that

lim P{Wn>x} = lim P(Mn<F'"^(x/n)} = e'^^, x>0, n—»oo n—»oo

since F'^(x/n) is a sequence of normalized levels. This means that the distribution of Wn is asymptoticaly exponential. Using this fact we have

nP{X>Mn}

= nP{nF(X)<Wi,}

=J

nP{nF(X)<x}dF^^(x)

-too

0= E(Wn)

and the result follows.*

In our simulation procedures the estimate of 0 was

computed based on levels determined in

accordance to the above described algorithm.

At a first step, sequences of size n=1000 of the maxautorregressive process {Xi} with a=0.5, 1.0 and

c=0.1, 0.5, 0.9 (i.e., 9 = 0.05, 0.1, 0.29, 0.5, 0.8, 0.9) were generated and estimates of 9 were computed for rn= 5, 10, 15, 20, 25, 40, 50, 100, 200, 250. This

procedure was repeated 500 times and, for each pair

(a,c) and for each rn,the average and the mean

square error of the corresponding estimates of the

extremal index were computed.

In fig.2, r^ is plotted against the estimates of E(9n)

for the different values of 0 considered above. From

its observation we can conclude that for r^ ranging

between 40 and 100 this procedure seems to work

reasonably well for all values of 0. Notice in

particular, the apparent agreement with the latter remark about r^ being significantly bigger than

knThe mean square errors represented in figure 3, do

not contradict the good behaviour of these estimates

for values of r^ in the same region.

In order to examine the empirical probability

distribution of the number of exceedances of the

random levels U-ai> which we expect to be

approximately geometric, another simulation

procedvire that generates sequences of n=1000 of {Xj}, was made for the same pairs (a,c). This procedure was run 1000 times and the empirical distribution of

Z, its sample mean and its sample variation were computed for different choices of r^ . Some of these

results are presented in table 1. Once again the

simulation resvdts suggest that the behaviour of this estimator is more sensitive to the ratio between the

size of each block and the total number of blocks than

to the value of 0.

109
Figure 2. Estimates of E(en), for n=1000 and

rn = 5,10,15,20,25,50,100,200,250.

8«Q.5

Figure 3. Estimates of MSECOn), for n=1000 and

rn = 5,10,15,20,25,50,100,200,250.

Table 1. Estimates of EOn), E(Z) and var(Z),

for n=1000 and rn = 5,10,15,20,25,50,100,200,300,400,500,1000

ca

e

ECZ) V»HZ)

rn 5 10 15 2) 25 50 IDO 200 300 400 500 1000

0.90 ao5 0.0B6 0.055 0053 0053 0.053 0/150

0.5 19.5 36017.72 18.30 3131&71

340

1882

358

1869

377

20.01 440

0.90 0.1 an 0.106 0103 am oaoi 0.102 ooss LO 10

90

9.4 78.6

9.4 83.7

9.7 839.8 889.9 101

9.75

96.34

20.2

108.2 0.50 029 0.29 0.29 03 03 0.29 029 0-3 0299 0.29

0.5 3.45 3.43 3.46 3.37 331 339 3.40 3.36 335 345

8.24 8.16 856 7.61 7.56 8.03 8.51 9.17 8.86 9.84 0.50 050 0.49 0.49 0.49 05/ OJO OJO OJO OJO 0.505 0.495 LO 2 2.1 2.1 20 20 20 2.0 20 20 1.98 2.0 2 2.0 2.1 2.1 2.1 2.1 2.0 2.1 Z3 2.2 2.4 0.10 oes 0.64 0.66 0.66 0.67 068 0.69 0.68 0.69 0.69 0.70 a67

0.5 1.46 1.56 1.52 1.52 1.50 1.48 1.44 1.46 1.46 1.45 1.44 1.48

0.68 0.86 0.71 0.74 0.72 0.69 0.62 0.67 0.69 0.75 0.69 0.84 0.10 O90 0.71 0^ 0^ 037 OS? oso OSl OSO OSO OSO OSl 0.^9 LO 1.11 1.40 1.22 1.16 1.15 1.14 1.11 1.11 1.10 1.11 1.11 1.11 1.12 0.124 0.63 0.30 020 0.17 0.16 0.16 0.11 0.125 0.12 0.14 0.13 0.16

References:

[1] Leadbetter, M.R. and Nandagopalan, S., On

Exceedance Point Processes for Stationary

Sequences under Mild Oscillation Restrictions. In: Extreme Value Theory. (eds.:J. Hiisler and R.-D.

Reiss), Springer-Verlag, 1989, p.69-80.

[2] Leadbetter, M.R. et al, Extremes and Related

Properties of Random Sequences and Processes,

Springer Verlag, New-York, 1983.

[3] Hsing, T. et al. On the Exceedance Point Process

for a Stationary Sequence, Prob. Th. Rel. Fields, 78,

1988, 97-112.

[4] Nandagopalan, S., Multivariate Extremes and

Estimation of the Extremal Index, Ph. D. Thesis.

Technical Report N°315, Center for Stochastic

Processes, University of Norh Carolina, 1990.

[5] Hsing, T., Estimating the Parameters of Rare

Events. Preprint, Texas A&M University, ,1990.

[6] Gnedenko, B.V., Kolmogorov, A.N., Limit

Distributions for Sums of Independent Random

Variables, Addison-Wesley, Cambridge, 1954.

[7] Cramer, H., Mathematical Methods of Statistics,

Princeton Univ. Press, Princeton, 1946.

[8] Alpuim, M.T., An Extremal Markovian

Sequence. J.Appl.Probab., 26, 1989, 219-232.

110
Approximate Extreme Value Analysis For A

Rigid Block Under Seismic Excitation

Facchini, L. and Spinelli, P.

University of Florence, Florence, Italy

The problem of the collapse risk (due to earthquakes) of a rigid block resting on a rigid

foundation is dealt with in this work; and three h\potheses are introduced to describe the series of

the seismic events:

1. the arrival rates of the single earthquakes during the structural life can be described by means

of a countmg process (specificalK , a Poisson random process);

2. the yearK' maxima for the peak acceleration of the soil can be described by a Fisher-Tvppet II distribution;

3. the single event can be described by a random non stationary' process. The behavior of the rigid block has been investigated via the statistical linearization method,

which can provide satisfactory approximations of the response of the system.

By means of a reasonable combination of the model for the seismic events and the approximation of the peak rotation of the block, an approximation was obtained of the p.d.f for

the extreme value of the rotation of the block due to earthquakes during a given period of time.

This is of crucial interest in the evaluation of the collapse probability of the block, as it can

conveniently be combined with matenal resistance distribution; the collapse mechanism for such a system involves two distinct characteristics: the former is the overturning condition, and the latter concerns the resistance of the block material (that is, the matenal can collapse before the

overturning condition is reached, thus causing the structural failure).

Introduction

negative 0 means rocking about comer 0 or comer

0' respectively.

The mechanical system taken into consideration is sketched in fig. 1: it is a rigid block free to rock

without sliding on either the base comers; its foundation is a rigid horizontal plane which moves in the x-direction according to a given function of time

Xg{t) which, in this particular case, will be assumed

to be a realization of the random process X„ ( / )

.

The equations of motion about each one of the base comers [5] are:

(p-(p + l = /(t)

(p-(p-l = /(x)

2.1

These equations describe the evolution of the system

via the lagrangian coordinate (p = 0/0^^ , where 0^^

angle measuring the tilting of the block; positive or

is the critical toppling angle (see fig. 1) and / is a

function of non-dimensional time parameter t;

111
besides, the non dimensional time parameter x = \it was introduced, where [i is the frequency of small

oscillations of the block, when it is suspended from

one of the base comers: specifically.

^MgR

I

2.2

O

where M is the mass of the block, g is the gravity

acceleration and Iq is the moment of inertia about

the comer 0.

fig. 1

Impact occurs between the block and its foundation

whenever there is a transition from rocking about one comer to rocking about the other; the associated

energy loss is accounted for by reducing the angular

velocity- of the block after impact.

Specifically, it is assumed that

Urn <^{t J

e- Urn i^(x) 2.3

where e is defined as the coefficient of restitution of

the angular velocity, and and x' are respectively

the nondimensional time parameters just after and

just before impact.

The statistical linearization of the system

The two equations of motion (2.1) and the linking

condition (2.3) can be conveniently summarized into the expression

q)+(l-e)6((p)(f)" (p" -(p + 5/^2((|)) = /(t) 3.1

Here, 6() is the Dirac Delta ftinction.

It is clear that the given system, even when it is slender enough and the equations of motion can be

piecewise linearized, is still strongly non-linear.

When /(i) is a deterministic fimction of time, the

behavior of the block can be investigated easily

enough because a closed form solution can generally

be obtained for each half cycle; on the other hand, if /(x) is a realization of a random process, it is generally necessary to make use of approximate

methods, such as statistical linearization [1] [4] is.

Before proceeding, we briefly discuss the

assumptions on the forcing process /(i)

.

Following Kanai-Tajimi [3] [6] theory, the power

spectral denity of the baseline excitation was

assumed:

1

i2

3.2

where ^ = 0.60 and co ^ = Stt; Sq is tlie intensity

function of a non stationary white noise obtained

from a Gaussian stationary white noise multiplied by

a deterministic ftinction of (actual) time %{t): in

particular, x{t) is made up by:

1 . a quadratic build-up:

(;^(/) =z t^/\6 for / < /| =4 sees.):

2. a constant term equal to unity (for

4 secs.= /] < / < /2 = 15 sees.);

3. an exponential decay:

ix(t) = exp(-0.0992{t-t2))

for 15 sees.= ^2 < 0 (fig- 2).

It can be shown that the power spectmm of the

forcing process f(T) is linked to Kanai-Tajimi model

by means of the relation:

Thus, one has to investigate the response of a linear

system whose equation of motion is

112
where

3.4

3.3

71 a

and Gcj)^ and c^j, are respectively the standard

deviations of the two processes O^, and .

Kanai - Tajimi model

2

1.2

0.8 -

0.6 -

0.4 -

0.2 -

10 20

frequency co

Envelope function

fig. 2

It has been assumed that, when x{t) = 1, i.e. from

/] = 4 sees, to t2 = \S sees, (or at least during a

portion of this time period), both the excitation and

the system response could be regarded as if they were

stationary. This is a rough assumption, and studies are at present being carried out to evaluate its adequacy. However, numerical results fi^om [8] (see

ne.xt paragraph), suggest it yelds satisfacton.' results. As usual, the variances of the two processes, tilt angle and angular velocity, were estimated via the

following relations:

3.6a

oc 6b

where

(co) = - 5/7/7(0)

CO 2 2

Extreme value distribution for a single earthquake

Once the excitation intensit\' is given via the

parameter Sq, the extreme value distribution during

the time period 4 secs.< / < 15 sees., i.e. when the

response is assimied to be stationary, can be

evaluated for the equivalent linear system by means

of Vamnarcke's [7] relation

- exp

1 - exp

'7: q^e

^

2 a

e J

exp

^ -2 ^

(P.

2o

-1

4.1

where:

1 ^O, \ M

2k a

4.2a

0'^2

113
00 4.2b

The variable ap does not affect directly ; Xj and

q are functions of -^OgO^ therefore of Op

.

By means of relation 4.1. it has been possible to check the validit\' of the approximations introduced

by the equivalent linearization; some numerical

resuhs were available from [8].

Reference of [8] studied the behavior of a rigid block

with aspect ratio 7? =10 ft. and slendemess

H/B-5: it was subjected to several ground

accelerations histories which followed the KanaiTajimi model. The mean peak acceleration was

a, OAg.

The coefficient of restitution was varied fi"om ^ = 0.90 to = 0.95; twenty- histories for each one

of the values of e were numencally integrated, and

the empirical cumulative distributions were

evaluated.

Fig. 3 shows the comparison between the empirical

fi-equencies obtained in Ref. [8] (the crosses) and the curve obtained by Vanmarcke's relation (4.1) for the

equivalent linear systems (computed according to the

proposed formulas) for four values of e (continuous

line); the dashed lines are the boundarv' of the

confidence mterval of a Kolmogorov-Smiraov

goodness-of-fit test.

Since each cross is contained in the confidence

interval, this gives evidence that the approximation

obtained was satisfactor)-.

Then, as the values of e did not differ much from

each other, all the empirical c.d.f were consolidated

so as to obtain a sample population of 120 samples: an equivalent linear system was computed imposing

e -0. 925 (that is, the mean of the previous values

taken for e).

Fig. 4 shows the extreme value distribution of

together with the empirical frequencies and the

confidence interval of a Kolmogorov Smimov test.

EXTR. DISTR. (e = 0.90)

"1 1—I

I I I I I

1.0

max EXTR. DISTR. (e = 0.91)

0.5 -

0.0

1.0

"1 f I I 1 r r I

0.1 1 max EXTR. DISTR. (e = 0.92)

-I I I 10

0.5 _

0.0

1.0

\ \ \ / \

1 U>7 1 1 1

\ A J

/ : y^ . . .

': aX\ \

~ / A '

\

/ ^ \/ A ^/ I i i / i ; ; / : :

1 L 1 1 1 U L .1.1 1 1 JUL

0.1 1^ max

EXTR. DISTR. (e = 0.93)

10

0.5 _

0.0

1 I —

1

1—r-1 1 IX-F» 1 1 -I -f-r 1 iT

/

/; .^y. i

; i / : Lx /

;/ /T ; I /\\\

- ^T'^V

.±/.

V /

/ i y /

, y /

/

1 11 l_J 1 t i.t . J .. -1- .J 1L-l—

1

-1-IL

0.1 10 max

fig. 3

114
(D

' max

fig. 4

Extreme value distribution

during the structural life

In the previous section the method for obtaining the

cumulative distribution function for the extreme

rotation of the block has been described. The starting

point was the shape of the power spectral density of

the forcing process, coupled to the mean of its peak

acceleration.

If an upper bound is imposed for the extreme rotation

of the block, for instance the toppling condition

|q)| < (p = 1, a safet\' region can be determined and

the curve v^'hich expresses the collapse probability

conditional on a given event having occurred can be

plotted.

In order for the seismic vulnerability to be estimated,

the collapse probability conditional on a given event having occurred must be combined with the

probabilitv' that this event takes place during the

structure's life. The upper bound for the rotation will be denoted by

(p and 5 = [-(p;(p] gR will stand for the safety

region of the dynamic system.

Let &s(^\cip) be the safety likelihood and

^c(^\^p) collapse probability for the block

115
subjected to a seismic motion whose mean peak

^((p^ a ) =

acceleration is On' ^ Py

= Prob Up <q) 5.1 We will assume that the time series of the seismic

events can be modeled as a Poisson process and that

the safet>' probability of the system does not change

with time: then the safet\' probability of the system

during its life T^^y is [2]

- exp' 5.2

where is the probabiht\' distribution function of

'py the yearly maximum of mean peak acceleration of

seismic motion for the considered site. From eqs. 4.1 and 5.1

5.3

Once the probability density function (p.d.f) of

yearly peak acceleration is given, eq. 5.2 may be

viewed as the probability of the rotation being less than an arbitrarily fixed value cp

^ dunng the system

life 7^^.; from this new point of view. eq. 3.1 defines

the cumulative probability function for the extreme

rotation during the whole structural life.

This new fiinction takes into account the random

structure of the seismic events in the considered site by means of the p.d.f of yearly maxima of ground

acceleration Pa(<^pv) '-

= exp 9 5 '^/?V )dapy

- Pr ob[<^ < g] 5.4

'py/\^Apy\^py,

during the whole structural life Tgj-j.. The corresponding p.d.f can be obtained by a

derivation of this last equation; it can be found that

-T,^exp

oc Tstr\A^s^'^py)'^py

^^j[(^,,apy)dapy 5.5 c(p^

Applications

The proposed method was used to evaluate the p.d.f

of the angle of rotation of Foca's column in Rome

dunng a period 7^^^ = 50 years ; the structure can

be roughly sketched as a 12.83 mt. cylindrical rigid

block whose medium diameter is approximately 1.27

mt (see fig. 5).

It shows a cntical angle 0^^ = 0.0984 rads, and a

natural frequency |^ = 1.069 s'^

Several forcing processes were considered (following

Kanai-Tajimi model with mean peak accelerations

ranging from 0.00 Ig to 1.000 g) and an equivalent

linear system was computed for each of them; in fig.

6 the extreme value c.d.f's are reported for ten processes, whose mean peak acceleration ranges from

O.lgto l.Og.

1.27 m.

12.83 m.

where

fig. 5

116
fig. 7

117
Extr. c.d.r during Tstr

0.02 0.04 0.06 0.08

This results were combined with the probabilitydistribution function of the yearly maxuiium of mean

peak acceleration of the ground; a Fisher-Typpet II type c.d.f. for tlie yearly peak acceleration was used:

Papy "1

( \

ll2 -1

exp<

."2

x-3

Extr. p.d.f. during Tstr

fig. 8

where Mj= 6.6552 10 and U2-\-^^\^ (see

fig. 7). The p.d.f. for the extreme rotation of the column was

computed (see fig. 8) by means of equations 5.4 and

5.5 assuming Tg^j- = 50 years.

Conclusions

A method to evaluate the extreme value distribution

for the response of a SDOF non linear system was

proposed: this method makes a massive use of the

statistical linearization technique, and is based on the

assimiption that both the excitation and the response

are stationary for a certain period of time, even though they are not. An approximation of the extreme value distribution

of the response of a nonlinear s\'stem during a given

time period (in our case, 50 years) was obtained.

0 0.02 0.04 0.06 0.08 0.1

References

[1]L. Facchini, Strutture monolitiche sottoposte ad

azioni aleatorie: applicazione al comportamento

sismico di opere monumentali. Graduate Thesis,

Univ. of Florence. 1990.

[2]V. Gusella. Structural failure and stochastic

discrete process of random events: an application

to historic building seismic vulnerability analysis.

Dept. of Civil Engineering report, Univ. of

Florence, 1990.

[3JK. Kanai, An empirical formula for the spectrum

of strong earthquake motions. Bull. Earthquake

Research Inst.. Univ. of Toky o. 39, 1961.

[4J J. B. Roberts. P. D. Spanos, Random Vibration &

Statistical Lineanzation, John Wiley & Sons,

1990.

[5J P. D. Spanos, A. S. Koh, Rocking of rigid blocks

due to harmonic shakmg. Journal of Engineering

Mechamcs, Vol. 110, No. 11, November 1984,

pages 1627 - 1642.

[6] H. Tajimi, A statistical method of determining the maximum response of a building during an

earthquake, Proc. 2nd World Conf. Earthquake

Eng., vol. II, Tokyo & Kyoto, 781-798, 1960.

[7] E. H. Vanmarcke, On the distribution of the first- passage time for normal stationary random

processes, J. Appl. Mechanics, 42, 1975.

[8]C.-S. Yim, A. K. Chopra, J. Penzien, rocking

response of rigid blocks to earthquakes.

Earthquake Engineering & Structural Dynamics,

8, 565-587, 1980.

118
The Rate Of Convergence Or Divergence For

Percentiles Of Gamma Distributions And Its

Application To Sample Extremes

Gan, G. and Bain, L.J.

University of Missouri, Rolla, MO

For a sequence (a ,B ) of positive constants with q 4-0 and B as n n n "^n ln(a ) ri^.^ ^^iM

ntoo, it is shown that if lim -a>0 , then lim 7-3—5—r -a -7—, nTcc ln(^^) nTco '71-^^(^2.1^2) ^2

where rj (^,k) is the lOOpth percentile of the gamma distribution with

mean and variance k^^, and if lim -7— - a>0 , then lim \—r =0

ntco Pn ntco (^2,^7)

1/k if ki<k2 , - «> if k2<ki, and - a ' 81/62 if ki=k2. An example of its

application to sample extremes is given.

1. INTRODUCTION

It is well known that the normalizing

constants to any domain of attraction

are all related to the percentiles of

the underlying distribution, and that

the gamma distribution belongs to the

domain of attraction of the exponential

type for the maximum and is attracted

to the Weibull distribution for the

minimiim. Also, by Ref. [1], the moments

of normalized extremes from the gamma

distribution tend to the moments of the

appropriate limiting distribution, and

so the approximation for the moments of

sample extremes based on the limiting

extreme value distribution is considered.

If one is interested in asymptotic

expectations of sample extremes from

gamma distributions and their ratio,

the rate of convergence or divergence

for percentiles of gamma distributions

will be helpful. This paper will derive

the rate in Section 2, and give an ex- ample of its application to sample

extremes in Section 3

.

119
Throughout this paper, rj^(8 ,V.) , 6>0

and k>0, will denote the lOOpth percentile of the gamma distribution Gam(^,k)

whose density function is given by

f(x:^,k) - k-1 -x/e X e•, x>0,

r(k)^'

and distribution function is denoted by

F(x;^,k), "x-^ViCx)" will represent a function V of x, and "f(x)iO (f(x)tO)

as x-»oo" will represent that J" is

strictly decreasing (increasing) and

lim f(x) -0.

2. MAIN RESULT

LEMMA 1 Let ( ctj^ ) he a sequence of

positive constants with cij^-'-O as nta>. Then

(1)

and

(2)

'1-a (^i,k)

lim

nn

lim

0<ki<k2

0<k2<ki

0<k2-ki

PROOF Since a -*n. n '1-Q ( ^ , k) is decreas

n

ing and differentiable for Q^e]0,l[, by

L'Hospital's rule,

n n

lim '-^ a^40 -^2ln[l-F(r?^.^(^i,k);^i,k)] $2 n

and (1) holds because a -10 as nt«. Next n

for Q^e]0,l[, since

(3) a^- Flr,^ (^.,k.);^.,k-], i-1 , 2 , n and (^£.k^) is increasing and

n differentiable

,

1 - f[n, (^i.k.);^.,k.]

n

da_ 'a ^ 1 ' i'^

' n n

i-1,2. Hence, for q^€]0,1[,

n n

n n

r(ki)^i kj-ki

k7 t'?a (^2.^2)]

r(k2)^2 n

•exp[} n (^i.ki)- i r? (^2.^2)].

Finally, (2) holds from a backward ap'

plication of L'Hospital's rule.

THEOREM 1 Let i<^^<0^) he a sequence of

positive constants with q iO and 10

In(a^)

as ntco and lim X^^Tg-y a>0. Then for

ki ,k2>0,

(4)

nt« "^l-B (^2.^2) ^ ^2

'^n

PROOF Since lim —t-t =a 7

ntco ^1-fl (^2-1) ^2

'^n follows from (1)

.

a 7-. (4)

LEMMA 2. Let a: ]0,oo[-]0,l[ and ^:]0,oo[

-»]0,1[ both be differentiable functions

120
with a(x)iO and /9(x)iO as xtco and

a(x)

lim

jgYxJ ' ^^'^ ctj^~<3:(n) and

^(n) for n-1,2 Then

(5)

ntco ^0 (^2,k) n

PROOF Refer to formula (3). ^a(x) -

f[r7^(^)(^:.k);^,,k] ^ r?^(^)(^i,k) and

^^f[.^(^)(^..k);^,.k]^,^(^^(^,,k).

d^^^^'^^ 1 1

•''a(x)(^-^)^ 7k !|

!i iil!a(xil!iilill!

1 a^t'?^(x)^^2,k)]

, and

(5) holds from a forward and a backward

application of L' Hospital's rule.

By applying (2) and (5), the following theorem holds and the proof is

omitted.

period of n days. A company has r meters, and some of r meters will be

charged at a discount rate if kept

separate, but r meters will be charged

at the same rate if pooled. If X^j represents the power demand for the j th

meter on the ith day, and -X^^+. . .+

X.^. then S(^)- max{S, S^}< X^^^^

+ . . .+ X(^)^, where X^^^^- max(X^j,

. . . ,X^j ) . So the question is when will

the reduction in the peak demand obtained by pooling meters offset the

added cost incurred by pooling. We need

an indication of the relative effect of

maximizing a sum compared to sums of

maxima. This information should be use- ful in other types of applications as well. For example, a greater peak load

capacity per work station is required

if the work stations are kept separate,

than if they are pooled.

THEOREM 2. Let (q ,6 ) be defined as in n"^n

Lemma 2 . Then

lim n

'^n

0 0<ki<k2

CO 0<k2<ki

a^/^^i/^s 0<k2=ki=k

3. AN EXAMPLE

EXAMPLE Suppose an electrical company's charges are based on the maximum

power demanded over a time

There are many potential applications

where analogous results for minima

would also be useful, so both cases are discussed below for the gamma distribution .

The gamma distribution is a flexible

two-parameter model, and results for

the gamma distribution should provide a guide as to what effects might

121
generally be expected. By applying Theorems 1 and 2 , the

following result holds

Let Xij ,

. . . ,X^j , j -=l,...,r, be r independent random samples selected from

Gam(^,k), X^^^j- the ith smallest order

statistic of X^j,...,X^j, ^^(\^) - j

E[X(^)j], and 6^(k) - ^E[X^^^.]. Then

the relative effect on maxima is given

by R (rk,k) - —7—TTT , and the relative

effect on minima is given by Q (rk,k) -

r5 (k)

^—(Tk)"- approximations for R^(rk,k)

and Q^(rk,k) based on the limiting ex- treme value distribution are given by

R^(rk,k) = R^Crk.k) -

T(2rk) + (l-7)x' i(2rk) 1-— 1--

ne n r[jx^ A2k)+ (l'y)x^ i(2k)] 1-— 1--

ne n and

• 2^2

Q^(rk,k) = Q;(rk,k) -

(2k) Tip

X^ (2rk) r(-{)

1/n

where 7=0.5772157 denotes Euler's con- stant and XpCv) denotes the lOOpth

percentile of the Chi-square distribution with V degrees of freedom.

lim R'(rk,k)

nt« n - and lim Q' (rk.k) - 0 r ntoo ^ which provides information for R^(rk,k)

and Q^(rk,k) when n is large.

REFERENCES

[1] Pickands, J., Moment Convergence of

Sample Extremes, Ann. Math. Statist.,

39 (1968), 881-889.

[2] Bain, L. J. and Can, G. , Comparison

of Expectations of the Extremes of Sums

and the Sum of Extremes from Gamma

Distributions, J. Statist. Comput . Simul. , 47 (1993) , 219-225.

It can be seen numerically from Ref.

[2] that the respective approximations

for R (rk.k) and Q (rk,k) based on

n n R^(rk,k) and Q^(rk,k) are surprisingly

good.

122
Application Of Extreme-Value Theory To

Reliability Physics Of Electronic Parts (On-Orbit

Single Event Phenomena)

Goka, T.

National Space Development Agency of Japan, Tokyo, Japan

Some models, for example weakest link model and bundle of fiber model used to be applied to reliability

physics in the fields of reliability and destructive engineering. To the test data and the statistical analysis of

the distribution of the smallest and the largest values that can be explained with these models, extreme-value

theory (particularly doubly exponential distribution) can be applied. The purpose of this paper is to examine

the application of extreme-value theory to the on-orbit data on single event phenomena of memory IC under

the space radiation environment. The application of extreme-value theory is compared with that of the conventional Poisson distributions to verify the effectiveness of the application of extreme-Value theory

(doubly exponential distribution).

l.INTRODUCTION

Since the destruction of material is thought to take place

at the weakest point of the component (the weakest link model), the material property has stochastic characteristics. Therefore, to correctly determine the tensile strength and/or

the life of a material, the distribution of smallest values is

appropriate. On tiie other hand, the distribution of largest values is significant to tiie leak current failure of electronic

parts or the corrosion pit depth for the rupture strength. The

selection of the smallest or the largest values of subject

data, in other words, will approach the extreme-value

distribution. Therefore analytical methods have been

established on the basis of extreme-value theory.

At present, a method based on reliability physics (a method applied to developing high reliability parts that are

completely fault-free by thoroughly pursuing the cause of

actually generated faults, detecting and correcting latent

faults through accelerated test) is drawing attention as the

ultimate decisive factor for reliability assurance. Under the circumstances, there are high possibilities for utilizing

extreme-value distributions, particularly the doubly

exponential distribution, in the field of reliability data

analysis of electronic parts. Single event phenomena are the well known interactions between high-energy particles in the space environment and

electronic devices on spacecraft. These phenomena are caused by high energy proton or heavier panicles such as helium, carbon, nitrogen, oxygen, iron and so forth in

galactic cosmic rays, the trapped Van Allen belt particles

and solar flares. These phenomena can be classified into Single Event Upset (SEU) and Single Event Latchup (SEL).

123
SEU is a reversible soft error that the information (digit 1 or 0) which are maintained in the memory or the

microprocessor unit (MPU) of the spacecraft are upset (1-*

0, 0-*l) by the particles in space. Especially, the upset

caused by protons is termed as proton upset, and that occurs even in the low-altitude orbit owing to the protons trapped

by the magnetic field of the earth. These protons are most

abundant above the south-Atlantic ocean, so called SouthAtlantic Anomaly(SAA).

SEL is an irreversible hard error which is caused by the high-energy particles to the electronic parts of the CMOS

technology.

2 MEASUREMENT OF SEU AND SEL BY ETSV

2.1 RAM SOFT-ERROR MONITOR

EXPERIMENT

Engineering Test SatelUte-V (ETS-V) has been launched

by NASDA on August 27,1987 and has been put into geostationary orbit at 150° east longitude. This spacecraft has a Technical Data Acquisition Equipment (TEDA) aiming at obtaining technical data which is necessary to develop

spacecraft. TEDA includes a RAM Soft-error Monitor

(RSM) that makes a measurement of the SEU or SEL

occurring at eight 64-kbit CMOS static RAM devices (NEC,

jj. PD4464D-20). The thickness of the shield of these

devices is estimated to be 21.5mm in Al. This monitor was

developed by NASDA in collaboration with NTT, Japan.[l]

September, 1989 and the 4B-class solar flares were

observed on September 29 and October 1989. The number

of SEL drastically increased during these solar Flares. The

number of SEU (sum of 8 devices) measured by RSM each

week is also plotted in Figure 2. From this figure one can

see that the number of SEU is less than that of SEL, and it

also increased remarkably when the solar flare occurred. [2]

200

160

^120

g 80

40

00 — r^. r- 000 600 1200 1800 DAY (The day of the origin=Sep. 1,1987)

Figure 1 Measured Latchup rate as a Function of Tirae(ETS-V)

100

80

60

UJ

40

20

00 i J .. .A* 000 6O0 1200 1300

DAY(The day of the origin=Sep. 1.1987)

Figure 2 Measured Upset rate as a Function of Time(ETS-V)

2.2 SEU AND SEL DATA ON ETS-V

(1) IN-ORBIT MONITORING RESULTS

Figure 1 shows the SEL data (sum of 8 devices) acquired

by the ETS-V TEDA RSM. The abscissa and the ordinate

axis in this figure indicate passing days and the number of

SEL that occurred in a week respectively. The period of data

acquisition is about 4 years from November 22,1987 to

Jan. 3 1,1 992. The solar activity became intensive since

2.3 POISSON DISTRIBUTION

The distribution of the ft-equency versus the number of

SEL and SEU in a week are plotted in Figure 3 and Figure 4

respectively.

The average number of SEL and SEU in a week are ML=2.4(/week) and MU=0.76(/week) respectively. If the phenomena are perfectly random and uniform process, these

distributions will agree to the Poisson distribution. In Figure

3 and Figure 4 dotted lines are the results of substitution of

these values in the formula of the Poisson distribution.

124
60

£5u

>

o

z

UJo

m 20

UJ

UJ>

o2

UJD

O

UJ

IT

V \\ \

\ \

\ \

/

10 NUMBEROF LATCHUP IN A WEEK

Figure 3 The Distribution of the Frequency Versus

the number of SEL(ETS-V)

120

90

60

30 \

10 NUMBER OF UPSET IN A WEEK

Figure 4 The Distribution of the Frequency Versus

- the number of SEU(ETS-V)

P(K) =

K!

(1)

where,

K:The number of SEL or SEU

P(K):The probability of observation of K SELs or K

SEUs

M:The average number of SEL or SEU (ML or MU,

namely)

These figures disclose that the observed data do not fit

the Poisson distribution very well. This is owing to a change

in the tendency of data before and after the solar flare which

occurred in the latter half of the total observation period.

Figure 5 and Figure 6 indicate the disdibution in the first (September 1 , 1987 to June 1 1 , 1989 : Solar Minimum) and

latter harf (June 11 , 1989 to January 31 , 1992 ; Solar

Maximum) of the total period as to SEL and SEU

respectively.

Figure 5 and Figure 6 indicate the distribution of the

frequency in the first and latter half of the total period as to number of SEL and SEU respectively.There are good

agreements between the data and the Poisson distributions

in these figures,and the number of SEL and SEU decreased

apparentiy after the solar flare.

60

The Latter Half Period

2 4 6 8 NUMBER OP LATCHUP IN A WEEK

Figure 5 The Distribution of the Frequency Versus

the number of SEL (ETS-V)

100 c ao>-

T

The Latter HaTf Period

x

Tb© First HalJ Period

2 4 6 8 10 NUMBER OF UPSET W A WEEK

Figure 6 The Distribution of the Frequency Versus

the number of SEU (ETS-V)

The data during the solar flares, namely August

12,September 29,October 19,1989 and May 21-25,1990 are removed from these figures.lt is apparent that the Poisson

distribution is inapplicable to the data when the solar flare

occurred. According to Figure 5-Figure 6,the number of SEL was

about 3 times as much as that of SEU.Generally,when SEL

and SEU occured simultaneously,only SEL will be detected by RSM because SEL is a hard-error.This means tiiat the

condition for the detection of SEU is that SEL should not

125
occur at the same time. Namely,

The condition for the detection of SEL:

LL<L

The condition for the detection of SEU:

LU<L<LL

where L is the LET (Linear Energy Transfer) of the incident

particle, LL is the threshold LET for SEL and LU is the threshold LET for SEU of the RAM devices. It is expected

that SEU would be comparatively hard to observe when the value of LU is close to that of LL.

2.4 DOUBLY EXPONENTIAL DISTRIBUTION

(EXTREME-VALUE THEORY)

Extreme-value theory is introduced to analyze the maximum single event rate data inclusive of data during

solar flares. Suppose we have a random sample from a

probability density function which has a tail that decreases

as exponential type (Poisson, normal, log normal, logistic,

etc.) and we are interested in the upper tail of the probability

density function (largest extreme values). The distribution function and probability density function of the so-called Type-I asymptotic distribution of largest values (double

exponential distribution) are, respectively,

F(y)=exp(-exp(-y)), ( 2

)

f(y)=exp(-y-exp(-y)). ( 3 )

Where y=(x- X)l a ,-co<x< oo,.oo< ^ < oo, q >o, and

location parameter X and scale parameter a are unknown.

Taking the natural logarithm of the distribution function

twice, we have

-ln(-lnF(x))=(x-A )/c (4)

Which stands for the equation of a straight line on the extreme-value probability paper. Assuming that single event phenomena comply with the Poisson distribution, the

distribution of the maximum values of the number of single

events will agree with the doubly exponential distribution. The cumulative probability of the maximum single event

rates (events/week) in a month are plotted in the left side of

Figure 7 (extreme-value probability paper). For this analysis, the data during the solar flare namely

August 12,September 29,October 19,1989 and May 21-

25,1990 are included to these figures. Figure 7 indicate the

distribution in the first half (1 .7 years) and the latter half (2.3

years) of the total observation period. The distributions become linear for the first half period, while the slope of the

line changes for the latter half period. Apparently this discrepancy depends on the effect of the solar flares.

3 SEU DATA ON MARINE OBSERVATION

SATELLITE-l(MOS-l)

3.1 SEU DATA ON MARINE OBSERVATION

SATELLITE-l(MOS-l)

Marine Observation Satellite- l(MOS-l) was launched by

NASDA on February 19,1987. Figure 8 shows the number

of SEU that occured in a day at the stored command memory

used in the command decoder of MOS- 1 . The period of data

acquisition is about 4 year and a half from September 1,1987

10 [

1 ! ;

8

-

000 BOO 1200 DAY(The day of the origin -September 1,1987)

Figure 8 Measured Upset rate as a Function of

to January 31,1992.

This memory consists of three bi-polar Static Random

Access Momory(SRAM) devices(93419, 64 9bit SRAM,

Fairchild). The shield thickness is assumed to be about

2.7nmi in Al. Though MOS-1 is a Low Earth Orbit(LEO,

909km) altitude and 99 degree inclination), one can see

apparently from this figure that the number of SEU (9 upsets

a day) increased by the effect of solar flare on October 19,

1989. The upset bits data are reported each second to the

ground. From this information the place where the upset has

occurred can be determined. Figure 9 is the upset map, in

126
PROBABILITY(%)

127
which the places where upset occurred from Feburary 1987

to August 1988 are indicated. This figure shows that the upset places concentrated on the so called South-Adantic

Anomaly.

Figure 10 shows geomagnetic field contour map on the MOS-I orbit spherical surface using IGRF85. One can see from Figure 9 and Figure 10 MOS-1 SEU occurred mainly

the region which corresponds to the area less than 20000

nano Tesla (geomagnetic total intensity) contour. A few

upsets occurred at both polar regions which correspond to about 70* north latitude and 70' south latitude. Ground

testing of the SRAM was carried out using high energy

proton at the cyclotron facility. Comparison of predicted

upset rate using NASA AP-8 model and orbital data was

carried out. There is quite good correlation between AP-8

predictioj^ and orbital data. [31

-90 -40 -30 0 30 Figure 9 Upset Map For Three 934 19's On MOS- 1 Spacecraft

CEonncNETic FiELO (NnNOiESLflS) or icnfos

TEfln- IS>87.0 nLT> 907.0 ( Kd

)

Figure 10 Geomagnetic Field Contour Map On MOS-1 Orbit

3. 2 STATISTICAL ANALYSIS OF MOS-1 SEU

DATA

(1) POISSON DISTRIBUTION

The distribution of the frequency versus the number of

SEU in a day during the total period is plotted in Figure 1 1. The distribution of Uie frequency versus the number of SEU

in a day during the first half period of the total period

(September 1,1987 to June ll,1989:Solar Minimum) and

during the latter half period (June 11,1989 to January 31,

1992:Solar Maximum) are plotted in Figure 12 and Figure

13 resp)ectively. These average number of SEU in a day are MU =1.13, MU2=0.82 respectively.

In Figure 12 and Figure 13 dotted lines are the results of

substitution of these values in the formula of the Poisson

distribution. These are good agreements between the data

and the Poisson distributions in these figures. The number of

SEU decreased slightly in the solar maximum period even

>

<

2. 600 >

o

z

tuO

UJ

iru.

400

200

0 2 4 6 8 10 NUMBER OF UPSET IN A DAY

Fig. 1 l.The Distribution of the Frequency Versus the Number

of SEU ( Total period )

500

400

300 ^ 200

-100

000

(M

1.13(1/0AY

OS-1)

B 0 2 4 6 NUMBER OF UPSET IN A DAY

Figure 12 The Distribution of the Frequency Versus the Number of SEU (First half period)

10

128
500

400

300

=> 200

100

000

v\\ Ml

/J=0.02(1/OA

MOS-1)

Y)

0 2 4 6 8 10 NUMBER OF UPSET IN A DAY

Figure 13 The Distribution of the Frequency Versus the Number of SEU (Latter half period)

for the MOS-1 spacecraft.

(2) DOUBLY EXPONENTIAL DISTRIBUTION

(EXTREME - VALUE THEORY)

The cumulative probability of the maximum single event

rates (events/week ) in a month are plotted in the center part

of Figure 7 ( extreme - value probability paper ).

3.3 SEU DATA ON THE TRACKING AND DATA

RELAY SATELLITE(TDRS-l) [4]

The TDRS-1 was designed by NASA, and was launched

from the space shuttle. Challenger in April 1983, and was

put into a geostationary orbit in July 1983. The SEUs were

observed in the Attitude Control System (ACS). The ACS

contains four pages of RAM, 256 bytes per page. Each page

consists of two static bi-polar Fairchild 93L422(256k X

4bit)RAM chips.

The TDRS-1 weekly SEU count shown in Figure 14 begins in April 1986. Some SEUs go unobserved. The

spikes in August, September and October 1989 are responses to solar flares. The off-scale responses in September and October are for weekly SEU total of 88 and

157 respectively. [4] The cumulative probability of the maximum single event rates (event/week) in a month of the TDRS-1 SEU

data are plotted in the right side of Figure 7 (extreme-value

probability paper) even for MOS-1 SEU, ETS-V SEU and

SEL. Figure 7 indicates the distribution in the solar

Minimum and the Solar-Maximum observation period.

Solar Minimum corresponds to the period of MOS-1;

September 1987-November 1988, ETS-V; November 1987-

November 1988, TDRS-1; May 1986- November 1988.

Solar Maximum corresponds to the period; November

1988- January 1991 in all satellites. The distributions

become linear for the Solar-Minimum period, while the

slope of the line changes for the Solar-Maximum period.

Apparently this discrepancy depends on the effect of the

1866 isgi

Figure 14 The SEUs showing that the envelope of the TDRS-1 SEUs clearly follows the modulation of the galactic cosmic

rays. The smoothed line through the TDRS data was created by using a cubic spline function . The spikes in September and

October 1989 reach 88 and 157 SEUs per week respectively. (Wilkinson, 1991)

129
solar flares. From all of ther analysis a decrease of the number of single events could be found during the

SolarMaximum. This phenomena are due to the screen

effect, that is to say the solar flare particles screen heavy

ions from the Galactic cosmic ray.

4. CONCLUSIONS

Upset, IEEE Trans. Nucl. Sci. NS-36 PP2344-2348, 1989

[4] Wilkinson D, Daughtridge S,Stone J, Sauer H, Darling

P, TDRS-1 Single Event Upset and the Effect of the Space

Environment, ffiEE Trans. Nucl. Sci. Vol 38 No. 6, PP

1708- 1712; 1991

Application of extreme-value theory (doubly

expotential distribution of the largest values) to the observation data of Single event phenomena about 4 years by the geostationary satellite and the medium altitude

satellite are examined.

In comparison with the Poisson distribution, the doubly

exponential distribution has the advantage to be able to analyze the data include the big solar flare and to enable us

to discriminate clearly from the effect of solar flares. A judgement from above point comes to the conclusion

that the doubly exponential distribution is preferable to the Poisson distribution.

5. ACKNOWLEDGMENTS

The author wish to acknowledge Dr.Shigeo Kase,

University of the Air for the helpful discussions concerning

extreme value theory, and the support of Mr.Kazuhiro

Kiyono for the analyzing of the data. The work described in this paper was carried out at the Tsukuba Space Center, NASDA.

6. REFERENCES

[1] Shiono N, Sakagawa Y, Sekiguchi M, Sato K, Sugai I,

Hattori T, Hirao Y, Single Event Effects in High Density

CMOS SRAMs, IEEE Trans. Nucl. Sci. NS-33PP1632-

1636, 1986

*

[2] Goka T, Kuboyama S, Shimano Y, Kawanishi T, The

On-Orbit Measurements of Single Event Phenomena by

ETS-V Spacecraft, IEEE Trans. Nucl. Sci. NS-38 PP1693-

1699, 1991

[3] Shimano Y, Goka T, Kuboyama S, Kawai K, Kanai T,

Takami Y, The Measurements and the Prediction of Proton

130
Certain Identities In Expectations Of Functions

Of Order Statistics And Characterization Of

Distributions

Govindarajulu, Z.

University of Kentucky, Lexington, KY

A method of generating identities in expectations of functions of order statistics defined on

the positive real axis is obtained. These identities are specialized to the exponential, the

folded normal, the folded logistic and the uniform distributions. The specialized identities are used to characterize the exponential, the generalized truncated normal, the folded logistic

and the uniform distributions.

Key words order statistics, identities, characterization of distributions.

AMS classification 62G30, 62E10.

1 Certain Identities in Expectations of Functions of order

Statistics.

Let X be a random variable having distribution

function F{x) and probability density function f{x)

where the latter is zero for x < 0. Let ^ be a continuous

difFerentiable function such that differentiation of g{x)

with respect ot its argument and expectation of g{x)

with respect to an absolutely continuous distribution

are interchangable. Furthermore let Xin < X2N <

... < Xnn denote the order statistics in a random

sample of size A'' drawn from f{x). Then we have the

following results. Proposition 1. If / is difFerentiable, then for

I < i < N and = 2, 3, ... we have

E[g'{XiN)] = -NmEg{Xi.i,N-i)

N

- J2 E[giXiN)f'{XkN)/f{XkN)]

.

k = l Proof. For 1 < i < N

E[giXiN+t)] = N\

J... J

0<2;i<...<rA/<oo .N g{xi + t)nj^J{xj)dxj = - /••/

t<yi<...<yt^<oo

9{yi)T^jLifiyj - t)dyj

/oo

131
where

yi<y2<<<yN<oo k=i ^ J{^kN))

Then

E[g'iXiN)] = j^E[g{XiN + i)]\t=o = -N\mh(0, 0)

-N\ / /'(yi)Myi,0)dyi + iV! /(l.l) Jo Jo f{yi)h'iyi,0)dyi (1.2)

/here

h{0

0<y2<yN<oo

9{yi)^^=2fiyj)dyj

-,Eg {N-l)\

{X^-l,N-l) (1.3)

/•CO poo N\ f'{yi)hiyi,0)dy^ = Nl f'{y,) Jo Joo

J J

9{yi)T^f=2f{yj )dyj

4/i<2/2<-<Vn<oo = E

(1.4)

5(^.7^)

/'(A'in)

rOO poo NU f{yi)h'iy,,0)dy, = N\ f{y,

J a Jo

I I

'<^'

Lyi<2/2<...<2/Ar<oo

-E f'iyk]

k=2

f'jXkN)

f{XkN) J (1.5)

Using (1.2) (1.3) and (1.4) in (1.1) we obtain the de- sired result. Remcirk 1. This method of deriving identities in expectations of functions of order statistics when the

support of the distribution is the real line ha^ been

used by Seal [10] and Govindarajulu ([5], p. 638).

Corollatry 1.1 Let g{x) = a; in Proposition 1. Then

we obtain

(1.6)

Special cases. Now if f{x) = exp(—x), for x > 0. Then (1.5) boromes

1 = N{E{X,n) - E{X,.,,N-i)} . (1.7)

If f{x) = {2/ny/^exp{-x^/2) for 0 < x < oo, then

for 1 < i < A?"

N

1 = -iV(2/7r)i/2^(Xi_i,;v-i) + Yl E{XiNXkN) . k=i

(1.8)

If fix) = 2e-^/(l+e-^)2 for a; > 0, then for 1< i < AT N

1 = -NE{X^.,,N-i) + J2 E{XiNF{XkN)} . (1-9)

k = l

Using analogous methods one can prove the following

proposition.

Proposition 2. If / is differentiable then we have

jv

fr[ j{XkN)

(1.10)

Remark 2. If Xq^n-i is taken to be zero, then

Proposition 2 can be included in Proposition 1. Corollary 2.1 Let g(x) = x. Then (1.9) becomes

N

k = l

f'{Xk,N)

f{Xk,N)

(1.11)

Special cases.

(a) Let f(x) = for x > 0 and zero elsewhere.

Then (1.10) becomes

l/N = EXi^N. (1.12)

(b) Let /(x) = (2/7r)i/2exp(-xV2) for x > 0. Then (1.10) takes the form of

JV

l = J2EiXi,NXk,N). (1.13)

it=i

(c) Let /(x) = 2e-^/(l + e-^)^ for x > 0.

132
Then since -f'{x)/f{x) = F{x) = {2/(1 -f e""")} - 1, (1.10) takes the form of

l = J2E{XiNFiXk,N)}. (1.14)

Remsirk 3. Since f'{x) = 0 the above propositi- ons are not applicable to the uniform density on (0, 1). In the following we obtain the specific results for the

uniform density.

Proposition 3. If / is the standard uniform density, for 1 < i < A'', and A'' = 2, 3 . .

.

E{g'{Xis)) = N{Eg{Xi,^^-^) - Eg{Xi.,^N-i)}

(1.15)

Proof. For 1 < i < N, consider

E[g{XiN+t)] = N\

J... J

0<Xi<...<X!^<l

g{xi +t)dxi . . .dxN = Nl

t<yi<...<yf^<l+t

9{yi)dyi --dyN

{i-iy.{N-iyl

{y-ty-\l+t-yf-'g{y)dy.

Thus

J'^\i-l){i-l)iy-ty-'il + t-yf-'

+{N - i){y - ty-\l + t - yf-'-'] g{y)dy

Hence

Eg'iXiN) =

{i-iy{N-iy

Jo

{{N - i)y - {i - 1)(1 - y)]dy

- iV! ~ {i-iy{N-i-iy

f g{y)y'-'{\-yf-'-'dy

Jo/ Q(v)y'~^ii-y)^~'dy. {i-2y{N -iy Jo ^^^'^ ^ '

Thus

Eg\XiN) = N [Eg{Xi^N-i) - Eg{Xi.i,N-i)]

Ki < N,N = 2,Z,...

Special Case 1 Let g{x) — x in (1.14) and obtain

1 = N[E{Xi,N-i) - E{Xi.i,N-i)] . (1.16)

By proceeding analogously we obtain Propositions 4 and 5.Proposition 4. If / is the standard uniform density, then for = 2, 3, . .

.

Eig'iXiN)) = NEg{X,,N-i) - NgiO) . (1.17)

Proposition 5. If / is the uniform density, then

for = 2, 3, . .

.

Eg'iXNN) = Ng{l) - NE{g{XN-i,N-i)} (1-18)

Special cases. Setting g{x) = a; in (1.16) and

(1.17) we have

and

AT - 1 N

l = E{X,,N-i),N = 2,d,

= E{Xn-i,n-i),N = 2,3,

(1.19)

(1.20)

Remark 4. If Xq^n-i is taken to be zero, then

Proposition 4 can be included in Proposition 3. If Xn.n-i is considered to be unity, then Propostion 5 can be included in Proposition 3.

2 Characterization of the Exponential Distribution

In this section we characterize the exponential dis- tribution using identities (1.6) and (1.11). Toward,

this we need the following notation. Let

Hiy) = infix : F{x)>u},0 < u < 1. (2.1)

Take F to be right continuous. Then for 0 < u < 1, we have

H{u) < X u < F{x).

Let Ui,...Un be a random sample from the

standard uniform distribution. Then the distribution

133
of H{Ui),H{U2),...,H{Un) is the same as that of

XuX2,...,Xn. A\soH{Uin)= min i/(C/fc) has the

^<k<N

same distribution as Xi^ etc. Throughout, we assume

that F is absolutely continuous. That is, H'{u) exists almost everywhere for 0 < u < 1. Then we have the

following propositions.

Proposition 6. Let F{A) = 0. Then for 2 < i <

iV and iV = 2, 3, . . . , E{XiN) - E{Xi.i,N-i) = l/N if and only if F{x) = 1 - exp(-(x - A)),x > A. Proof. From the proof of Proposition 1, we can

write

iV! ^(^^^) = (i-i)!(iV-i)!

Jo Then writing N = {N - i + 1) + (i - 1) we have

(A^ - 1)! E{Xi,N) - E{Xi.,,N-,) =

(._2),(;v-i)!

Jo {N - ly.jN - i + 1) ^ (i- 1)!(A^- 0!

Jo Performing integrating by parts in the first integral

by writing u'~'^du as d{u'~^/{i — 1)) we obtain (after

cancelling out terms and noting that H{0) = A)

Now writing

we have

E{Xi,N) - E{Xi-i,N-i) = for 2 < 1 < AT and

iV = 2,3,...

imply that

(.-'fjK^v'-o- i'"'"''^-"''"''^'"""-"-"^

u = 0 for 2 < i <

and N = 2,3, — Now the only continuous function

which is orthogonal to u'~^(l — u)^~\ a linear combination of u, . . .u^~^ iov N = 2,Z, . . . is the zero function itself. Hence H'{u){l — u) — I — 0 for almost all u in

(0, 1). Integrating on both sides we obtain H{u) + c =

-ln(l-u).

Now H{^) = A implies that c = -A. Also H'{u)

exists for all u in (0, 1). Now it follows that 1 — F{x) = exp{ — {x — A)), x >

A. Proceeding in an analogous manner one can prove

the following.

Proposition 7. Let F{A) = 0. Then for N =

1,2,...E{Xin) = A+l/N if and only if F(x) = 1 -

exp(-(x - A)), X > A. Remark 5. By defining Xq^n-i = A, Proposition 7 can be included in Proposition 6.

3 Characterization of the Generalized Truncated Normal

Distributions

In this section we characterize the folded normal

and the generalized truncated normal distributions,

using identities (1.7) and (1.12).

Proposition 8. If F(0) = 0, then for =

2,3,..., Y^k=i E{XiNXkN) = 1 if and only if F{x) =

2$(x) - 1 for 0 < X < oo. Proof. See Theorem 3.2 of Govindarajulu ([6], p. 1013) or Theorem 6 of Lin ([9], p. 403).

Proposition 9. If F{A) = 0, then for 2 < i <

iV - 1 and A^ = 2,3,...

Y,E{XiNXkN) = l + N{EX)EiXi.i,N-i)

k = l

if and only if

, $(x) -$(A) , ,

1 - $(A)

where $ denotes the standard normal distribution function.Proof. One can write

N j-1

^E{XiNXjN) = ^

j=l j=l

134
N

(i )!(iV-i)! J J - 2)!(iV

zwF'-^iw)[l - F{w)f-'dF{z)dF{w)

+

ii-l)\{N -iyJo

F'-^{z)[l - F{z)f-'dF{z)

+ 7T(i-l)\{N -i-iy. II

zwF'-\z)[l - F{z)f-'-^dF{t)dF{w)

.

After performing integration by parts with respect to w once in the first double integral we obtain

N

J2EiXiNXjN)=J— {i - iy{N - i^.11

z<w F'-\w)[l - Fiw)]'^-'dF{z)dw

+

- Nl

{i - iy.{N - i - 1)1 J J

zw{F'-\w)[l- F{w)f-'

z<w +F'-\z)[l - Fiz)f dF{z)dF{w)

(i- l)!(iV -"^.11

Z<UI zF'-\w)[l - F{w)f-'dF{z)dw

H-iy.iN-i-iy.j,

i! [l-F{w)]^-'-^dF{w)) . Now when / is the truncated normal density, the right

hand side simplifies to

1 - fiA)N{-fii-i,N-i + tii.N-i} + NE{X)fXi,N-i = 1 + NE{X)fii-i,N-i{since f{A) = EX)

.

|| On the other hand Yif=i ^i^iNXjN) = 1 +

NE{X)fii-i^N-i implies that

-7^!

{i-iy{N ZTiy j J

H{u)H'(v)v'-'

(1 - v)^-'dudv

+NE{X)ni^N-i = 1 + NE

{X)fii-i^N-i (3.1)

Since

dv

and

fii-i,N-i- (•_2)t(Ar

(A^-l)'

{l-v)^-'dv = {N-iy

{N

{N - iy.ii - ly

H{v){\-v)^-'d{v'-^)

{l-v)^-'dv+ -

{N-iy

So

(^"- l)!(iV-i-l)!

/ H{v)v'-^{l-v)^-'-^dv . Jo

Nifii^N-i - /it-i,;v-i)

(i-iyXN

Hence (3.1) can be written as m —J\'{v)v'-^{l-v)^-' dv

(i - iy.(N - 0!

/

v'~\l-v)^-'[-H'{v) f

H{u)du] + {EX)

Jo Jo

f

H'{v)v'-\l-v)^-'dv

Jof

v'-\l-v)^-'dv

Jo = 0,i = 2,...,N - l,N = 2,Z...,

That is, for all most all v in (0,1)

-H'{v) I

H{u)du + H'{v)EX -1 = 0; Jo

le

H'{v) I H{u)du = 1 since EX = f

H{u)du

Jv Jo 0<u<v<l (3.2)

135
Now, using the arguments in Govindarajulu ([6], p. 1012) we can establish that F(x) = {^(x) -

<^{A)}/{1 - ^A)} {or A<x <oo.

Proposition 10. If F{A) = 0, then for =

2,3,...

N

Y^EiXjNXNN) = 1 + NE{X)E{Xn-i,n-i)

if and only if F{x) = - $(A)}/{1 - $(A)} for A < X < oo. Proof

N-l

L.H.S=Y;^ +E{XU) = N{N-1)

I I

z j=lw

Z<VJ

F^-^{w)dF{z)dF{w) + E{X'^j^). (3.3)

Now

E{Xlr,) = N z / ——NF''-\w)dF{w)dF{z)

= N J J

zF^-^{w)dwdF{z)

w<z +N{N-l) J J

wzF^-'^{w)dF{z)dF{w).

w<z Now use this in (3.3) and combine the two symmetrical double integrals and proceed cis in the proof

of Proposition 9.

4 Characterization of the Folded Logistic Distribution

In this section we characterize the folded logistic

distribution using the identities in (1.8) and (1.13).

Proposition 11. Let F{A) = 0. Then

E{XFiX)} = 1 -

f

if

1 + F{x) = 2[1 + e(^-^)]-i - 1 for X < A .

Proof After performing integration by parts once we obtain

E{XF{X)} = J

xFdF = jF{l-F)dx

+ J

x(l-F)dF.

That is

2E{XF{X)} = JF{l-F)dx

+

J{l-F)dx + A

= J{2f-{1-F)}dx

+

Jil-F)

dx + A

= 2 + A.

Proposition 12. If F{A) = 0, then for 1 < f <

iV,iV = 2,3,...

ZLi E{Xi,NFiXk,N)} = ff;(Xi_x,jv-i) + 1 if and only if

1 + F{x) = 2(1 + e(^-^))-i ,x>A.

Proof. Consider

N

J2E[Xi,NF{Xk,N)]

k = l

1 = J2+ E{XiNF{XiN)}

k = l N

+ J2 E{Xi,NFiXk,N)}

i-l

k=i+l

{i-2y.N -i

+- zF'

iw)[l - F{w)]^-' F{z)dF{z)dF{w)

{i-l)l{N-i)lJo

{z)[l-Fiz)f-'dF{z)

^{i-l)\{N-i-l)\ J J

z<.w

(z)[l - F{z)r-'-'F{w)dF{z)dF{w)

.

We can write the last integral (without the constant multiplier) as

1 /"^ r

' wF\l- F)^-UF{w)^

a dFiw) = —

(i-l) J JzF'-\z){l - F)^-'Fiw)dF(w)

z<w

136
dF{z)+ J J

F'-\z)

z<w

(1 - F{z)f-' F{w)dF(w)dz

Thus

N

^E{XiNFiXk,N)} = -rr m

k = l {i-2y.{N-i)\

J J

{wF'~^{w)[l- F{w)f-'F{z)

+

^zF'-^{z)[l- F{z)f-^F{w)]

dF{z)dF{w)

{i-l)\{N-i)\ J J

'^'^ z<w

(1 - F{z))^-'F{w)dF{w)dz

2{i-2)\iN-iy.J^

(z)[l - F{z)f-'dFiz)

+: JA (z)[l-F(zr-{l-F2(z)]dz

N 1 jV r

= Y^(^.--i.iv-i) + 2{i-i)\{N-iy.

F'-\z)[l - F{z)f-'[1 - F^{z)]dz . Now for the logistic distribution on (yl,oo),/(2) =

|(1 — F^{z)): hence the last integral on the right

side reduces to unity. Now J2^=i F{^iNF{Xk,N)} —

^E{Xi-x,N-\) + 1 for 1 < i < TV and iV = 2,3, . .

.

imply that

cfu = 0, for 1 < i < TV and = 2, 3 That is, H'{u) = for almost all u in (0, 1). Integrating on both sides and using the fact that

fr(0) = A, we obtain

H{u)-A = ln{(H-u)/(l-u)} and H'{u) exists for all u .

Now letting u = F{z) and x = H{u), we obtain the

result.

We will give the following propositions without

proofs which are analogous to that of Proposition 12. Proposition 13. If F{A) = 0, then for N = 1,2,..., Y:k=iF{Xp,NFiXk,N)} = 1 +

yE{Xn-i^n-i) if and only if

F{x) = 2{1 + e(^-^)}-i - 1 for a: > ^

.

Proposition 14. If F(0) = 0, then

N

1 = E{XiNF{Xk,N)} for iV = 1, 2, ... if and only if

F{x) = 2(1 + e-^)-^ - 1 fori > 0 .

k = l

5 Characterization of the Uniform Distribution

In this section we characterize the uniform distri- bution using the identities in (1.15), (1-18) and (1.19).

Proposition 15. For 1 < f < A^, = 2, 3.

1 = JV[£(X,,jv-i - E{Xi.i,N-i)] if and only if F{x) = x,0 < X < I. Proof. Consider

Nl

N{E{Xi,N-i)-EiXi.i,N-i)}

• / H{uy-\l-uf-^du.

- 0' ^0 {i - 2)!(A^

Now

1- i-AH{u)u^-\l-uf-^\l

-

I H'{uy-\l - uf-'du

Jo -(i-l) I H{uy-^{l-u)^-'du]

Jo

du +

N-i

i- 1 N - i

I H(u)u'-\l-uf-'du

Jo

137
Hence

N{E{Xi,N-i)-E{Xi-i,N-i)}

L.HS= 1, for l<i<N,N

= 2,3,... imply that „ /' "^-'(1 - uf-' {H\u) -l]du

{i - \)\{N - i)\ Jo = 0{oT l<i< N,N = 2.3,... .

That is

H'iu) = 1 for almost all u in (0, 1) . Thus

H'(u) exists for all u in (0, l)and H'{u) = , . f{F~'-{u)

Hence = lor/(a:) = l

Proposition 16. F(0) = 0 and E{Xin) =

1 /{N + 1) for = 1, 2, ... if and only if F{x) = x,0 <

X <1.

Proposition 17. F(0) = 0 and E{Xnn) =

N/{N + 1) for iV = 1,2, .. . if and only if F{x) =

x,0< X <1.

Proofs of Propositions 16 and 17 are analogous to the proof of Proposition 15. (For Proposition 16, see also Galambos and Kotz ([4], pp. 55-57) or Lin ([9], Theorem 1, p. 398)).

Remcirk 6 Ahsanullah [1] has characterized the

uniform distribution using the identical distribution of

Xnn — Xi^N and Xn-i^n in the cl;>'>s of all super or subadditive and absolutely continuous distributions.

Note that a distribtuion F is said to be super (sub)

additive if F(a; + J/) > F{x) + F{y){F{x + y) < F(x) +

F{y)} for all x,y>0.

References

[1] Ahsanullah, M. On characterizations of the uniform distribution based on functions of order statistics. Aligarh Journal of SiaiisticsQ (1989) 1-6.

[2] Bennett, C. A. Asymptotic Properties of Ideal

Estimators Ph.D. Thesis, University of Michigan

(1952).

[3] Govindarajulu, Z. Exact lower moments of or- der statistics in samples from the chi-distribution

(l.d.f.) Ann. Math. Statist. 33 (1962) 1292-1305.

[4] Galambos, J. and Kotz, S. Characterizations of

Probability Distributions (1978) Lecture Notes in Mathematics, Springer, Berline-New York.

[5] Govindarajulu, Z. On moments of order statistics and quasi-ranges from normal populations. Ann.

Math. Statist. 34 (1963) 633-651.

[6] Govindarajulu, Z. Characterization of normal and

generalized truncated normal distributions using

order statistics Ann. Math Statist. 37 (1966)

101]-1U15.

[7] Govindarajulu, Z. Characterization of the exponential distribution using lovper moments of or- der statistics pp. 117-129 in A Modern Course on

Statistical Disl l ibutions in Scientific Work (1974)

(Editors G. P. I'atil, S. Kotz and J. K. Ord). D.

Reidel Piibli^li'mg Co. D'li.lrecht. Holland.

[8] Govindarajulu, Z. and Lindqvist, B.H. Asymptotic efficiency of the Spearman estimator and Characterizations of distributions. Ann. Inst. Statist. Math. 39 (1987) 349-361.

[9] Lin, G. D. The product moments of order statistics with applications to characterizatoins of dis- tributions. J. Statist. Planning and Inference 21

(1989). 395-406.

[10] Seal, K. C. On minimum variance among certain

linear functions of order statistics. Ann. Math.

Statist 27 (1956) 854-855.

138
Investigating The Bias And MSE Of Exceedance

Based Tail Estimators For The Cauchy

Distribution

Grimshaw, S.D.

Brigham Young University, Provo, UT

One approach to estimating the tails of the cumulative distribution function, quantile

function and probability density function is to use only those observations in the sample

which exceed a high threshold. This investigation for the Cauchy probability model will indicate how the threshold selection affects the bias and MSE of tail estimators.

Key words and phrases: Generalized Pareto distribution, Hill's estimator.

1. Introduction

Suppose that the possible observed values from

a population can be characterized by a random vari- able X whose probability model is estimated using a sample from the population. The properties of

this estimated probability model which correspond

to the population characteristics of interest are the

foundation of statistical analysis.

Three important functions of a probability

model for a continuous random variable are the absolutely continuous distribution function F{x) =

F[X < x], the quantUe function Q(u) = F~^{u) and

the density function given by f{x) which represents

P[a < X < 6] = jlf{x)dx for a < h. The signifi- cance of these three functions in statistical analysis

foUows from their interpretation as key properties

of the population.

This work focuses on the problem of estimating the tails of F{x), ^^nd /(x) from a random

sample. The sample (empirical) distribution function, sample (empirical) quantile function and non- parametric density estimates are typically used in early stages of statistical analysis v/hen minimal as- sumptions are made on the underlying probability

model. However, these estimators prove unsatisfactory for values in the tail since they are confined to

the observed sample values and ignore the possibility of more extreme values than the observed sample in future observations. For example, the empirical quantile function in the case of insurance claims

would not estimate any insurance claim larger those

already observed in the sample, that is, it assumes

the largest insurance claims have already been filed and any future insurance claims will not exceed the

sample extremes. This restriction to tail estimation

is unacceptable.

The classical approach to tail estimation is to assume the underlying probability model belongs to some known class V whose elements are indexed

by a parameter 6 taking values in a set 0, that is V = {Pg, 6 e 0}. The distribution function, quantile function and density function then have para- metric representations F{x;6), Q{u-9) and f{x:6).

Tail estimates are given by F{x: 0), Q{u; 0) and

f{x: 0), where 6 denotes an estimate of the parameter 0 based on the sample information. For example,

if the underlying probability model is assumed to

follow the normal probability law, then 6 = {pl,(t),

the mean and standard deviation, with estimator

6 = {X,s), the sample mean and sample standard

deviation. The tail estimator is formed by replacing

the unknown parameters with the estimators in the

normal distribution function, quantile function and

139
density function.

The beauty of this classical parametric ap- proach is tarnished by what Fisher [1] called the

problem of specification. Often it is difficult to select a single parametric family for the population. Several candidates may appear reasonable judging from

their fit to the observed values. To demonstrate

this complication, suppose that a random sample

is taken from a population characterized by a symmetric unimodal probability model. Two possible

parametric families are the normal and the Cauchy.

Inference on the central values of the random vari- able will be similar for either of these parametric

models. However, the focus of this work is on tail values, not central values, and inference at the tails

is quite different under the two parametric models. Extremely small and extremely large values are much more likely under the Cauchy modeUng. The

distribution function F{x) approaches zero and one much more rapidly under the normality assumption.

The quantile function Q{u) for the Cauchy model

decreases more rapidly in a neighborhood of zero and increases more rapidly in a neighborhood of one. The density function f{x) for the Cauchy model has

much more area in the tail.

It is very difficult to discriminate between the

different possible parameterizations even when the

possible parametric models specify very different

tail properties. Very large sample sizes are required

for a goodness of fit test to have sufficient power to detect differences in the observed tail and the fitted

tail under the normal and Cauchy modeling.

2. Exceedance Based Tail Estimators and

Their Properties

The main objective of this work is to obtain

estimators of F{x)^ Q{u) and f{x) which allow the

data, not the parametric family, to dictate the tail behavior of the underlying probabiUty model. These

estimators can be used in applications to validate

tail behavior properties in probability modeling applications. In this discussion, the upper tail is dis- cussed without loss of generality since the lower tail results follow immediately after one observes that

the lower tail becomes the upper tail if the data is negated.

Grimshaw [2] proposed the following paradigm

for tail estimation:

1. From a random sample Xi,. .

. , X„, choose, as a function of n, a threshold percentile tn = k/n

close to zero, for some integer k.

2. Estimate the corresponding threshold Tn =

X{n — k; n), the (n — ky^ order statstic.

3. Obtain parameter estimates (p, a) from the ex- ceedances Xi — Tn for all Xi > Tn4. Estimate the tails of the quantUe function, dis- tribution function, and density function by

Q'{u) = Tn +a -9

l-u

-P

tn

'

for 1 - *n < u < 1,

F'{x) = l-tn- g-^ (^-^\x-Tnl-^

for Tn<x< Q(l),

nx) = tn\-{g-')' {-\[x-TnY-'^

a V ^ /

for Tn<x< Q(l).

where

A2)i/\ A < 0, 2 < 0

a-^iz Al - <! A = 0, 2 < 0

-1/A < 2 < 0

and

' (l + Az)(i/^)-i, A<0, z<0

e^ A = 0, 2 < 0

(l + A2)(VA)-i, A>0,

-1/A < z < 0

Grimshaw [2] has shown that these estimators are asymptotically normal if the estimators (p, a) are asymptotically normal as —> co. Two options are available for estimating the

parameters {p,a). One approach suggested by

Pickands [3] is to model the k exceedances of the

threshold as a sample from a generalized Pareto

distribution (GPD) whose parameters can be esti- mated by maximum likelihood. One example of this approach is given by Smith [4] to estimate extreme

ozone levels. Grimshaw [5] has proposed an algorithm for computing the GPD maximum likelihood

estimates. Let (pgpd^^gpd) denote the maximum

likelihood estimates from the GPD model for the

exceedances.

140
A second approach was suggested by Hill [6] using

1 X{n — f + 1; n)

k

^

Xin — k\ n)

It has been shown by many authors that for these es- timators based on the exceedances of a high threshold, as —* oo,

PGPD

IS AN , (nt) ^V'gpd

where

and

where

^GPD =

PHill w AN

^

P

.flHill. a

-a(p-l-l) 2a2(p+l)

< (nt)-^VHili

ap a?

VHill =

3. Effect of Threshold Selection for the

Cauchy Distribution

This investigation focuses on the Cauchy probability model with

=^ -I- ^ tan"^ X, x G E. =tan7r(u - i), 0<u<l

According to the tail behavior classification of Parzen [7], the Cauchy distribution has a tail ex- ponent of /) = 1 and therefore is classified as a long

tailed distribution.

Figure 1 shows the bias of the proposed tail estimators for t = 0.25, 0.15, 0.10, 0.05 for the

distribution function, quantile function and density

function, respectively. As anticipated, as t 0 the

bias is reduced. For both the distribution function

and density function it is interesting to note that

the largest bias occurs immediately following the

threshold. Figures 2(a) and 2(b) compare the MSB

of the distribution function tail estimator for the

GPD estimators and HUl's estimators. Figures 3(a)

amd 3(b) compare the MSE of the quantile function

tail estimator for the GPD estimators and Hill's es- timators, and Figures 4(a) and 4(b) compare the

MSE of the density function tail estimator for the

GPD estimators and Hill's estimators. A sample

size such that nt = 30 is assumed in each of these

figures. In all cases it is clear that the superior precision of Hill's estimators translate into superior tail estimators.

From these figures, it is obvious that a small

value for t results in better estimates. However, the

evaluation of a choice of t must be based relative to the desired region for the tail estimate. For example,

the choice t = 0.25 provides an excellent estimator

of the quantile function for 0.75 < u < 0.95. But

if a more extreme tail estimate is desired, say from

0.90 < u < 0.98, then t = 0.10 is a more appropriate

choice.

A simulation study was performed to investi- gate the effect of threshold selection on tail parameter estimators. In this simulation, threshold values

of i = 0.25, 0.15, 0.10, 0.05, 0.01 were chosen with

the number of exceedences of the selected threshold

of nt = 30, 50, 100. Table I contains the estimated mean and estimated variance of the tail estimator

parameters for the Cauchy distribution based on 100

simulations.

Notice both the GPD and Hill estimators are biased for the tail parameter p = 1- This bias is not due to any failure of the methodology. Grimshaw

[2] showed that estimators of the tail exponent are asymptotically unbiased as nt oo, but the rate

of convergence can be very slow. Further, it can be

shown that for /? > 0 Hill's estimators are asymptotically superior. From the simulation, it appears

that Hill's estimators are superior in finite samples

and that for the Cauchy probability model the bias

appears quite small.

It appears from the simulation that threshold

selection affects the GPD and Hill estimators dif- ferently. For PgpDi as t decreases and nt increases

at each step the estimator improves incrementally.

In contrast, pHill has a large improvement changing

from t = .25 to t = .15, but the remaining choices of

t yield nearly equivalent estimators. Further, there

is little change in the bias as nt is increased. This

indicates that pHill niay be more robust to threshold

selection than pgpdAlso notice that the parameter a depends on

the choice of t. Grimshaw [2] showed that the pa- rameter a is a function of t and must be asymptotically equivalent to t/ fQ{l — f) as i ^ 0, where

fQ{u) denotes the density-quantile function. For

the Cauchy distribution, at i = 0.25, a ~ 1.571; at

141
t = 0.15, a ~ 2.286; at t = 0.10, a ~ 3.290; at

t = 0.05, a ~ 6.419; and at t = 0.01, a ~ 31.841. In estimating a it appears that dcpD provides a good

estimator for all choices of t but dniii only performs

well for small values of t and large values of nt. Therefore, it appears that Hill's estimator of p

is rather robust to threshold selection but the corre- sponding estimator of a is poor except as t —* 0 and

nt —* oo. The GPD estimator of p is inferior compared to Hill's estimator, but the pair (pgpd, ^opd)

demonstrate a steady incremental improvement over

the values of t and nt. REFERENCES

[1] Fisher, R.A., Statistical Methods for Research

Workers, OUver and Boyd, London, 1948.

[2] Grimshaw, S.D., "A unified approach to estimating tail behavior," Ph.D. Dissertation, Texas

A&M University, (1989).

[3] Pickands, J., "Statistical inference using extreme

order statistics," Annals of Statistics, 3 (1975)

119-131.

[4] Smith, R.L., "Extreme value analysis of environmental time series: An application to trend detection in ground-level ozone," Statistical Sci- ence, 4 (1989), 367-393.

[5] Grimshaw, S.D., "Computing maximum likeli- hood estimates for the generahzed Pareto dis- tribution," Technometrics, 35 (1993), 185-191.

[6] Hill. B.M., "A simple general approach to infer- ence about the taU of a distribution," Annals

of Statistics, 3 (1975), 1163-1174.

[7] Parzen, E.,"Nonparametric statistical data modeling," Journal of the American Statistical Association, 74 (1979), 105-131.

142
Figure 1. Bias of the Tail Estimators of F{x), Q{u) and f{x), respectively, for the Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

143
Figure 2(a). Mean Square Error of the Tail Estimators of F{x) using the GPD parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

Figure 2(h). Mean Square Error of the Tail Estimators of F{x) using the Hill parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

144
MSEQ(ul,.25)

MSEQ(u2,.15)

MSEQ(u3..10)

0.75 0.8 0.85 0.9 0.95

ul ,u2,u3

Figure 3(a). Mean Square Error of the Tail Estimators of Q{u) using the GPD parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

Figure 3(h). Mean Square Error of the Tail Estimators of Q{u) using the Hill parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

145
MSEf(xI..25)

MSEf( x2,.15)

MSEf( x3,.10)

1 1 1 1 1

'.1

'.\A - - - .

.

1

1 5 10 15 20 25 30

xl ,x2,x3

Figure 4((^)- Mean Square Error of the Tail Estimators of /(x) using the GPD parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

Figure 4(b)- Mean Square Error of the Tail Estimators of f{x) using the Hill parameter estimators for the

Cauchy Distribution for threshold percentiles t = 0.25, 0.15, 0.10.

146
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CM

O 00 C^ 1—1 o o o o 00 c lO 00 t—1

00 o to o CO 1—1 o CO CO CM o o 1—1 c lO lO c o c c o o o CO T—1 o 1—t o lO o o 1—1 lO CM o CO T-H CO o CM o 1—1

LO o 1—1 o 1—1 CM

lO o o CO o 00 LO

LO CO CO 1—1

CM o CM 1—1 00

p c CM o pp

1—1

CO

p o o 1—i 1—( c CO 1—1 o CO c 1—( o CO o QP

CO OO CM o o O CO 1—(

1—( o CO o 00 ^ o lO CO rCM r- 00 lO IV LO r- CM 00 00 o LO 00 LO lO o T—1

t-- lO 00 o CO t—1

00 o CM CO o IV

p .—( p

ppp 00 o CO c^ c CO CM c CO d d d^ CO d

oo" 00 1—1 00 CO CO cm" CO 1—f

1—1 CM CO CM r- CO 1—4 o 00 LO 1—1

CO LO CM o 00 LO LO o CO o CO c o o IV

CO LO CM o 00 to 1—(

00 00

p 1—1 COpp 1—1

T—1 p o p

p CM o d CM d 1—I d cm' d

o" CO cT o co" co' lO LO 1—<

LO 1—(

1—1 CO LO 1—1 o 1—1 LO r- oo IV OO CO 1—(

00 c o lO c LO c CM CO c CM o Ci 00 CO 1—( c LO CM o 1—(

CO OO

p LOp c CM CM c o CM o c CM d^ T—1 d CM d

OO r- CO CM LO o o 1—(

1—1 CM CO

CM 00 <y> 00 o 00 LO CO 1—1 IV

1—(

-^r 1—1 o CO CM 00 CO r~ o O 1—1 o LO c c CM CO co CM lO CO «D 1—(

CM

1—1 c 1—

H

1—1 1—1

p 1—1p 1—1 o 1—1 o

r-(

d 1—1d 1-5d 1—1d 1—( d 1—1

d

CM CM o lO

LO o 1—(

CT) CO 00 LO o 00 CO CO ^ o o o C2 lO CO

00 1—1 00 LO 1—1 IV CM

1—1 CO CO 00 CO 1—(

OO 1—1 LO CO 00 o 1—1 P 1—1 d d 1—1 d d d 1—1 d d d 1—1 d

C4. e oCO o

to

Si o

II

147

Estimating Quantiles For A Type III Domain Of

Attraction Based On The k Largest Observations

Hasofer, A.M.

The University of New South Wales, New South Wales Australia

Wang, J.Z.

University of Western Sydney, New South Wales, Australia

A method of estimating the high quantiles of a distribution belonging to the

domain of attraction of type III extreme value distribution (reversed

Weibull) is proposed by means of the W statistic, which is used to determine

the extreme value domain of attraction. The procedure is based on the k

largest order statistics from a sequence of n observations. The problem is treated by a three-parameter model and the endpoint of the distribution is estimated. A test of hypothesis is used to eliminate cases where die endpoint

does not exist. The estimators are shown to be asymptotically consistent.

Simulation results are provided.

1. Introduction and summary

Suppose a distribution F(x) is in the

domain of attraction of a distribution

H(x), which has been identified to be the

same, up to location and scale, as one of

the extreme value distributions, whose

types are given by

Type I (Gumbel):

H^(x) = exp{-exp{-x}}, oo>x>-oo; Type II (Frechet):

Hj^(x) = exp{-x'^}, x>0;

Type III (reversed Weibull):

a^yi^) = exp{-(-x)^}, x<0,

where y is some positive constant (see

Ref. [2]). The problem of estimating large

quantiles of the distribution has been

addressed by several authors.

In Ref. [6] an estimator for large

quantiles of a distribution based on the k

largest observations is derived. It is assumed that the distribution belongs to the domain of attraction of H (x) and the

derivation is based on the asymptotic

distribution of the k largest order

statistics of the sample as the sample

size goes to infinity. For distributions

in the domain of attraction of H (\) and

i,y

H Ax), however, there are three

2,y

parameters and therefore a different

approach is needed.

For the three-parameter problem

Hasofer and Wang (Ref. [4]) suggested

estimating high quantiles of a

distribution in the domain of attraction

of H ..(x) by a local maximum likelihood

1.y

method based on the extremal distribution

of the k largest order statistics. This

procedure works for all values of y, provided n, the sample size, and k are

large enough.

The three-parameter problem for

distributions in the domain of attraction

of H (x) was studied by Smith and

2,y

Weissman (Ref. [5]). The method is based on a local maximum estimation procedure.

However, the approach sometimes fails to yield an estimator because the likelihood

function fails to have a needed local

149
maximum, when y<1.

In this paper, we suggest that the

three-parameter problem for a distribution

in the domain of attraction of H ,,(x) can be solved by means of the W statistic. The

endpoint of the underlying distribution is estimated by solving a simple equation

based on the W statistic and the relation

between H^(x) and H^y(x). We shall show

that the estimator is asymptotically

consistent. The W statistic was proposed

by Hasofer and Wang (Ref. [3]) for testing

the extreme value domain of attraction.

The statistical properties of the W

statistic were studied and the asymptotic

distribution was determined.

2. The W statistic

The statistic W introduced by Hasofer

and Wang (Ref. [3]) is a function of the

top k order statistics of a sample of size

n: X > ••• > X , and is given by

In kn

W(X, , X ) =

k (X-X ) kn ^"^ X

^ (X -X)'

1 = 1 in

where X = (X X )/k. The critical values

1=1 in for the null hypodiesis that the

distribution of X is in the domain of

attraction of H^(x) are given in Table

VIII of Ref. [3]. It was shown there that a value of W lower than the lower critical

point indicates that the distribution

belongs to the domain of attraction of

Hj^(x), while a value higher than the

higher critical point indicates that it belongs to the domain of attraction of

H^^(x). It was also shown that under the

null hypothesis the W statistic is asymptotically normally distributed with

1 3 mean k and variance 4k' . The power of

the W test was studied by extensive

simulation.

3. Estimating the endpoint

Suppose that the distribution of X,

F(x), is determined to be in the domain of

attraction of H (x). Then the support of

2,y

F(x) must have a finite upper bound cOq, say. Let X, > ••• > X^ be the it top

In kn order statistics of a sample of size n. In

this case the limiting distribution of the

Y = -Mcon - X ), i = 1, k, is,

in "in

after a transformation of scale and origin

by a suitable pair of sequences, the

extremal distribution corresponding to a

distribution in the domain of attraction

of H^(x) (Ref. [6]). Note that the value

of W is invariant with respect to the

above linear transformation.

Our proposal for dealing with the

estimation problem for distributions in

the domain of attraction of H^,,(x) is to

2,7

A

seek an estimator of cOq, co, such that, for a given sample.

W

=e[w^(Uj,

X^;, -MO. - XJ

which is asymptotic to 1/k as k ^ <». Here

E(-) is the expected value and the U.'s

have the joint density

h^(Uj, Vi^ = exp|-exp{-U|^} -lu.j,

V.

i = 1 / for Uj> ... > u^. In the following

discussion, we denote X^,, by X^, i = 1, k, and

VK((0) = wj^-Mco - Xj), -Mo) - X|^)j.

The estimation of co enables one to make the transformation and then to estimate the quantile as for the case of a

distribution in the domain of attraction

of H^(x) (Ref. [6]).

3.1. The limits of W((xi)

We look at the limits of as a function of CO, when co tends to X^ and «».

Lemma 1. Suppose X^ > X^. Then

150
W^(co) = . (k-1)^

Proof. Let Y = -£n(co - X), i = 1,

1 1 k, and

G(co) = J- +

^

Then

G(co) =

k k(k-l)V^(a))

I ( Y.-Yj'

1 k

i = 1

, k - 1 y

Note that Y > Y . Since Y, «> and all

1 2

' Other Yj's remain finite, then

1 and

"^^1 (k-1)^

Lemma 2.

Proof. This can be shown by observing that ^

co(Y, - Y,) = X, - X„

for i = 1, k-1.

3.2. The monotonicity of V^(co)

We are going to show that V^((o) is an

increasing function for all co > X^ > ... > X . k Lemma 3. The function

f(^\ - (x-c)/(cx)

^^''^ £a(c/x)

is a strictly monotone increasing function

for all 0 < X < c.

Proof.

af(x) ^ x"^[£AC-£ax+((x-c)/c)]

[Mc/x)]^

By noting that

£ax = £^+((x-c)/c)-((x-c)^/2^^), 0<^<c,

the lemma can be directiy proved.

Theorem 1. U^(co) is a stricdy monotone

increasing function for all co > X^ > X , where the X 's are not all equal.

k 1

CO - X^

CO - X.

Proof. Let

0 < y = Y-Y = in

1 1 k

y[ = ayyaco, i = 1, k

aG(co)/5co = N/D

A -

1

and

Then

where

D = 2'

/« - 1 -v .

k - li - 1 N = I I (y.-y.)y.y.[(y'/y.)-(y'/y.)]

. ...» J » J 1 1 J J

1 = ij = 1 (see Ref. [3]). Note that

[(co-X.)-(co-X )]/[(co-X )(co-X )]

/y — * IK ' ' £a[(co-XJ/(co-X. )]

k 1 Then by lemma 3,

y7y. > y7y., for i > j, 11 J J since co-X > 0)-X. Moreover we have y <

1 j "'i

y., for i > j. Therefore N < 0. And, since

the X. are not all equal, then N < 0 and D

> 0. So aG(co)/5co < 0 and avy(co)/aa) > 0. Now it is clear that the proposed

estimator of cOq inevitably exists when the

W test rejects the null hypothesis at some

level less than 20%, say, in favor of

Hp^: the random variable is in the domain

of attraction of type III, since W(cl))

varies from l/(k-l)^, which is less than

Uk) to W{X,

which is greater than E V^(U,

for k > 3.

3.3. Simulation results

The simulation is based on the

limiting distribution of the k top order

statistics of a sample from H^^(x). The

joint density is given by

151
= V[(-Xj) ... (-x^#'exp{-(-x/}

for 0 > X ^ > x^ (see Ref. [6]). Here

we set k = 50 and

e[w^(U^, U^^)j = 1/50.

Figure 1 shows a successful case when

the estimator of cOq exists. As co increases

from X , the value of W increases over r

1/50. Figure 2 shows a failure case when

the value of W(X^, X^^) is lower than

1/50. VV^(co) increases slowly up to W(X^,

X ) as 0) tends to «>. 50Table 1 is a simulation result with

2000 replications on estimating endpoint

of h^y(Xj, x^^) with an endpoint COg = 0. N.L. is the number of samples for

which the solutions of fi) do not exist. The

differences in N.L. for each y are apparently because of the power of the W

test: with lower value of y the power of

the test is higher.

Table 2 is a simulation result with

2000 replications on estimating endpoint

of h ,Xx ,

COq = 0.

^100^ with the endpoint

0.030-

0.02(>

0.0

1

WiOi)

0.030-

0.020-

0.010-

Table 1

Y 0.6 0.8 1 2 3 4

EST I MATED

VALUES

MEAN -0.223 -0.177 -0.130 0.052 0.156 0.169

SD 4.195 2.077 1.535 1.354 1.334 1.313

N. L . 0 0 0 22 33 104

Table 2

Y 0.6 0.8 1 2 3 4

EST I MATED

VALUES

MEAN -0.313 -0.239 -0.144 -0.085 0.025 0.140

SD 3.819 2.014 1.416 0.807 0.954 1.144

N.L. 0 0 0 0 0 12

152
3.4. Consistency of the estimator of

endpoint.

Assume that X > ... > X have the

1 k joint density \y('^^^ \)' to

location and scale with the end-point cOq. We have the following.

Theorem 2. The proposed estimator is asymptotically consistent.

The proof is given in the appendix.

4. Estimating quantiles

Let the upper e-quantile, denoted by

q , of a random variable X be defined by

P(X < q^^) = 1 - e, where e is some positive small number.

Suppose that the distribution function of

X is in the domain of attraction of

H^^Cx) and the endpoint cOq has been

A

estimated with value co by the proposed

method based on the top k statistics from

a sample of size n. Then the estimator of

quantile is given by

A A rA A "\ q = CO - exp-^b - ^.tncy

where

A k -

and

b = ks^ - V +

with

Y. = -Mco - X.), i = 1, k,

s = i^-; i-^ . k 1=1 and = 0.5772-

(Ref. [6]).

As an example of estimation of

quantiles, simulation based on 1^(0,1) with

replication 3000 is carried out (Table 3).

The sample size is 500 and the top 55 is selected. In order to compare our method with

some classical methods, the LINT procedure

for percentile estimation (see Ref. [1])

is carried out. Table 4 is obtained by

simulation with 3(XX) replications.

Another example of estimating

quantiles by the proposed procedure is based on the reversed exponential

distribution with replication 3000 (Table

5). The sample size is 500 and the top 55

is selected.

Table 3

p 0.950 0.975 0.990 0.995 0.998

EXACT QUANTILE 0.950 0.975 0.990 0.995 0.998

ESTIMATED

VALUES

MEAN 0.949 0.975 0.990 0.994 0.997

SD 0.00866 0.00626 0.00369 0.00256 0.00216

Table 4

p 0.950 0.975 0.990 0.995 0.998

EXACT QUANTILE 0.950 0.975 0.990 0.995 0.998

ESTIMATED

VALUES

MEAN 0.948 0.973 0.988 0.993 0.996

SD 0.00989 0.00718 0.00487 0.00355 0.00276

153
Table 5

p 0.950 0.975 0.990 0.995 0.998

EXACT QUANT I LE -0.0513 -0.0253 -0.0101 -0.00501 -0.00200

ESTIMATED

VALUES

MEAN -0.0517 -0.0257 -0.0106 -0.00570 -0.00276

SD 0.00905 0.00636 0.00370 0.00258 0.00214

Appendix: Proof of Theorem 2

Note that the asymptotic solution of

VV'(CL))=l/k is the same as the asymptotic

solution of kG(a))=2 (see the proof of

Lemma 1). Write now

-a

and

h = (co-coo)k'"- > 0

-au: Xi = coo-e" where a = 1/Y and the U.'s have the joint

density

h^CUj u^) = exp|-exp{-u^} - I

for Uj > > u . Then

k -au; Yi = -£n(0) - Xi) = -£n(co - COq + e"^ 0

and

where Tj = Uj - U^. Following the same

steps as in Ref. [4] (proof of consistency

of a), we obtain diat

lim kG(0)) = f(h)/[g(h)]^ = C(h),

say, almost surely, where

-ax

f(h) =

J

e-'M^[^

o g(h) =

[

e'" In

+h

-ax +h

1+h

dx

dx.

Since V^((o) is a strictly monotone

increasing function of CO, kG(co) is a

strictiy monotone decreasing function, and

thus must be non-increasing. Since

^(0) = 2 and, as will be shown, C(0) < 0,

it follows that the equation ^(h) = 2 has

a unique solution: h = 0. Hence the unique

solution (ii of kG(w) = 2 will tend almost

surely to cOq. We now show that C(0) < 0 for all a >

0. By direct calculation we have the

following:

f(0) = 2a\

f(h) =

I

2[^Me'^+h)-Ml+h)J

^(e'^+h)''-(l-Hh)'^je"dx,

g(0) = -a,

g(h) = j"[^(e-"'+h)"'-(l+h)"'je"dx.

When 0 < a < 1,

f(0) = -2a[^(l-a)-^ '

g(0) = (1-a)-^ - 1, ^

r^Ax _ g'(0)f(0)-2f(0)g(0)g(0)

^^^^ ?(0) . = -2aW(l-a)^ < 0. When a = 1, we have that

C(h) ~ f(h) + 4g(h), as h ^ 0.

It is easy to show that

f(h) ~ -ilnhf+2itnh)(inil+h)), as h ^ 0,-

g(h) ~ -Inh, as h -> 0. Hence

C(0) = J^rp C(h) = -

When a > 1, aXih) ~ f(h) + 4ag(h).

154
It can be shown, by direct calculation,

that

f(h) ~ [2B+2h^"^^"Mh-2A£n(l+h"^)]/h^'^^",

as h -> 0, and

g(h) - A/h'-"«,

as h 0, where A and. B are some positive

constants. Hence

a\{h) " -2AM1 + l/h)]/h'"^^", as h ^ 0,

and

C(0) = J^iji C(h) = -~. -

References

[1] Boos, D. D., "Using extreme value

theory to estimate percentiles,"

Technometrics, Vol. 26, No. 1 (1984), 33-

39.

[2] Galambos, J., "The asymptotic theory

of order statistics" 2nd ed. Robert E.

Krieger Publishing Co., Malaber Fla, 1982.

[3] Hasofer, A. M. and Wang, Z., "A test for extreme value domain of attraction,"

J. Am. Stat. Ass. Vol. 87, No. 417 (1992),

171-177.

[4] Hasofer, A. M. and Wang, Z.,

Esdmating quantiles for a Type II domain

of attraction," (Submitted for

publication, 1992).

[5] Smith, R. L. and Weissman, I., "Maximum likelihood estimation of the

lower tail of a probability distribution,"

J. R. Statist. Soc. B, 47, No. 2 (1985),

285-298.

[6] Weissman, I., "Estimation of

parameters and large quantiles based on

the k largest observations," J. Am. Stat. Ass. Vol. 73, No. 364 (1978), 812-815.

155

Extreme Values Of Monotonic Functions And

Evaluation Of Catastrophic Flood Loss

Lambert, J.H., Li, D. and Haimes, Y.Y

University of Virginia, Charlotttesville, VA

A univariate monotonic function is often a useful model in engineering risk assessment, e.g., in relating the magnitude of flood discharge to the consequent economic losses. An approach is developed in this paper to determine the domain of attraction for a monotonic function of an underlying random

variable. Using von Mises' criteria, sufficient conditions are derived to find the domain of attraction of a transformed variable for situations where only incomplete knowledge concerning the underlying random

variable and the monotonic function is available. The sufficient conditions, along with relationships for the transformation of the extremal statistical parameters, lead to a practical methodology for estimating the expected loss, conditional on the exceedance of a threshold level or percentile. The conditional expected

value can serve as a measure of the risk of extreme events, such as catastrophic floods.

INTRODUCTION

In engineering assessment of the risk of extremes, a

univariate monotonic function is often a useful model

relating an underlying random variable and the

consequence. Knowledge of the tail of the probability

distribution of the underlying random variable can be

applied to derive the distribution of die system outcome. There is considerable literature on applications of the

statistics of extremes in engineering, including Refs. [1]-

[4]. However, it is difficult to quantify the risk associated

with the extreme outcomes when the distribution of the underlying variable is uncertain due to scarce data and lack of knowledge of the physical process. Using stadstics of

extremes, this paper studies the tail of the distribudon of a monotonic function of an underlying random variable

whose exact form of probability distribudon is unknown.

Background

Let fx and Fx be the probability density function and

cumulative distribution function, respectively, of an

underlying random variable X. The largest sample value,

X ^ , from n independent observations of X is itself a random variable with the following cumulative

distribution function

Fxniax(x) = [Fx(x)]n (1) n As n approaches to infinity, the distribution of the largest

sample value from X usually converges to one of the three

particular forms, or domains of attraction: the Gumbel,

which is of a double exponential form; the Frechet, which

is of an exponential form; and the Weibull, which is of an

exponential form wiUi an upper bound (Refs. [2], [3], and

[5]). There exist in the literature (Refs. [3]-[6]) various

forms of necessary and sufficient conditions for determining the domain of attraction of X. A simple set of

sufficient conditions for determining the form of the asymptotic distribution is von Mises' criteria (Refs. [2] and [7]). From (Ref. [2]), the distribution of the largest value

from X converges to a Gumbel form if X is unlimited in the direction of the largest value and

lim d l-Fx(x)

x-^ dx fx(x) = 0 (2)

In the Appendix of this paper it is shown that Eq. (2)

implies that

X->oo X IXW

and that Eq. (3) implies Eq. (2) if the function (1 - Fx)/fx

is a monotone function (increasing or decreasing) for large

X. The monotonicity condition is satisfied by most, if not

all, common distributions in the Gumbel domain of

attraction. Following Ref. [2], we consider in this paper

only distributions of Gumbel forms that are unlimited in the direction of the largest value. From (Ref. [2]), the distribution of the largest value

from X converges to a Frechet form if X is unlimited in the direction of the largest value and there exists a strictly

positive constant k such that

x—>oo 1 - Fx(x)

157
From (Ref. [2]), the distribution of the largest value

from X converges to a WeibuU form if X has a finite upper bound w which satisfies w = sup{x : Fx(x) < 1 } and

there exists a strictly positive constant k such that lim (w-x)fx(x)^^

j^^Q x^w 1 - Fx(x)

Two important parameters in the statistics of extremes

are the characteristic largest value and the inverse measure

of dispersion. For the underlying random variable X, the X

characteristic largest value, u , is defined in (Ref. [2]) by

Fx(u^)=l-^ (6)

and the inverse measure of dispersion, 5 , is defined in

(Ref. [2]) by

5 n = n fx(u5

or equivalently,

-X dln(n)

(7)

(8)

Ref. [2] shows that for many distributions of engineering

interest, the characteristic largest value and inverse measure

of dispersion as defined here are sufficient to parameterize

each of the three extremal forms.

The hazard function fx(x)/[l - Fx(x)] (Ref. [2]) is equal to the corresponding inverse measure of dispersion

X X

5_(x), where n is determined by Eq. (6) with u equal to

Objectives of the Paper

A strictly monotone increasing function represents

situations in which the higher the realization of the underlying random variable, the higher the outcome. A

function, Y = g(X), of an underlying random variable X is considered in the following, where g is assumed to be a

strictly monotone increasing function. It is often possible

from the observation data to determine for X a domain of

X

'n' attraction, the characteristic largest value, u and the X

inverse measure of dispersion, 5^^, by means such as

Gumbel's extremal probability paper and the method of

moments or the method of order statistics (Refs. [2] and

[3]). However, the exact form* of the underlying

distribution of X is probably never certain. The problem is to study the extremes of the function Y with only limited

knowledge of the probabilistic description of X. In this direction, Ref. [3] demonstrates conditions under which a monotone transformation preserves the Weibull asymptote

in the smallest value. Sufficient conditions are derived in this paper to identify for the random variable Y its domain of attraction

based on the von Mises criteria. Expressions are also

Y

obtained for the characteristic largest value, u^, and the Y

inverse measure of dispersion, 6^. The derivation does not

assume or require knowledge of the specific form of the

distribution of X. These results are then used to evaluate a

conditional expected value that is a measure of the risk associated with extreme events. The organization of the paper is as follows. Results in the form of nine sets of

mapping routes and conditions are first derived that identify the domain of attraction of the function Y. Expressions are then developed to derive the characteristic largest value and

the inverse measure of dispersion for the function Y. These

results are used to assist in the calculation of the estimate

of a conditional expected value of Y in the extremal range. An application of the developed method in assessing

catastrophic flood losses is presented.

PRESERVATION AND TRANSFORMATION

OF DOMAIN OF ATTRACTION

Preservation and Transformation of the Gumbel Domain of

Attraction Assume the distribution of the largest value from the

initial variate X to be of the Gumbel asymptotic form.

Sufficient conditions are derived here under which the asymptotic distribution of the largest value from Y is preserved in the Gumbel asymptotic form or is transformed

to the forms of Frechet or Weibull.

Theorem 1: Assume Y = g(X) is unlimited in the

direction of the largest value. The asymptotic distribution

for the largest value of the function Y = g(X) is of a X

Gumbel form if (i) [l/6jj(x)] [dg(x)/dx]/g(x) is monotone

for large x and its limit is 0 as x approaches infinity, or

(ii) the limit of d[lng(x)]/d(lnx) is finite as x approaches

X

infinity, X satisfies the condition in Eq. (3), and [l/5j^(x)]

[dg(x)/dx]/g(x) is monotone for large x. Proof: We consider convergence criterion of the

Gumbel domain of attraction in Eq. (3) for Y:

lim 1 - FY(y)

y^oo yfY(y)

lim l-Fx[g-ky)]

y-^yfX[g-^(y)]dg-l(y)/dy

lim 1 - Fx(x)

x->oo g(x) fx(x) dx/dg(x)

lim 1 - Fx(x) X dg(x)/dx

x^oo xfx(x) g(x)

(9)

The above limit is equal to zero if either the limit of

X

[l/5j^(x)] [dg(x)/dx]/g(x) is 0 as X approaches infinity, or if

d[lng(x)]/d(lnx) is finite as x approaches infinity when X

satisfies the condition in Eq. (3). Then it can be concluded

from Eq. (3) that the asymptotic distribution of the largest

158
value of the function Y is of a Gumbel form if [l/5jj(x)]

[dg(x)/dx]/g(x) is monotone for large x. 0

Theorem 2: Assume Y = g(X) is unlimited in the

direction of the largest value. The asymptotic distribution for the largest value of the function Y = g(X) is of a X

Frechet form if the limit of 5 ^^Cx) g(x)/[dg(x)/dx] exists

and is stiicUy positive as x approaches infinity. Proof: We consider the von Mises convergence criterion of the Frechet domain of attraction for Y:

lim y fY(y)

y-^ 1 - FY(y)

lim y fx[g~-^(y)] dg-^(y)/dy

y-x« i-Fx[g-ky)]

^

lim fX(x) g(x)

x^oo 1 - Fx(x) dg(x)/dx

If g(x)/[dg(x)/dx] and [1 - Fx(x)]/fx(x) are of the same

order of magnitude as x approaches infinity, then the Umit

in the last expression is strictly positive and it can be

concluded fi-om Eq. (4) that the asymptotic distribution of

the largest value of the function Y is of a Frechet fomi. 0

Remark. Note that if X sausfies the convergence

criterion of the Gumbel form in Eq. (3), a necessary condition for Y = g(X) to satisfy the von Mises

convergence criterion of Frechet is that the limit of

d(lnx)/d[lng(x)] exists and is equal to 0 as x approaches

infinity.

Theorem 3: The asymptotic distribution for the largest

value of the function Y = g(X) is of a Weibull form if there exists a finite (O such that co = sup{y: FyCy) < 1 } and

X

the limit of 5 jj(x)[co - g(x)]/[dg(x)/dx] exists and is strictiy

positive as x approaches infinity. Proof: We consider the von Mises convergence criterion of the Weibull domain of attraction for Y: lim (CO - y)fY(y)

y_>Q) 1 _ FyCy)

lim (0) - y)fx[g~^(y)]dg-%)/dy

y-^O) l-Fx[g-l(y)]

1- ^^[co-g(x)]^ x->oo 1 - Fx(x)

CO - g(x)

dg(x)

(11)

^~dg(^^ and [1 - Fx(x)]/fx(x) are of die same order of

magnitude, then the limit in the last expression is strictly

positive and we conclude from Eq. (5) that the largest

value from Y is asymptotically of the Weibull form. 0

Remark. Note that if X satisfies the convergence

criterion of the Gumbel form in Eq. (3), a necessary condition for Y = g(X) to satisfy the von Mises

convergence criterion of the Weibull form is that the limit of [CO - g(x)]/{x[dg(x)/dx]} exists and is equal to zero as x approaches infinity.

Preservation and Transformation of Frechet Domain of

Attraction Assume the distribution of the largest value from the

initial variable X to be of the Frechet form and that Fx

satisfies the von Mises convergence condition in Eq. (4). Sufficient conditions are derived here under which the asymptotic distribution of the largest value from Y is preserved as Frechet or is transformed to Gumbel or Weibull.

Theorem 4: Assume Y = g(X) is unlimited in the

direction of the largest value. The asymptotic distribution for the largest value of the function Y = g(X) is of a Gumbel form if the limit of d[ln g(x)]/d(ln x) exists and is equal to zero as x approaches infinity and

X

[ 1/5 ^

(x)] [dg(x)/dx]/g(x) is monotone for large x.

Proof: We consider the convergence criterion of the Gumbel domain of attraction in Eq. (3) for Y:

lim 1 - FY(y)

y_>oo yfY(y)

lim l-Fxlg-^y)]

y^yfx[g-ky)]dg-l(y)/dy

lim 1 -- Fx(x)

x->oo g(x) fx(x) dx/dg(x)

1 lim x dg(x)/dx

k x—>oo g(x)

(12)

When the limit of d[ln g(x)]/d(ln x) is equal to zero as x approaches infinity, from Eq. (3) the distribution of the

largest value from Y = g(X) is asymptotically of a Gumbel

X

form if [l/6j^(x)][dg(x)/dx]/g(x) is monotone for large x. 0

Theorem 5: Assume Y = g(X) is unlimited in the

direction of the largest value. The asymptotic distribution for the largest value of the function Y = g(X) is of a

Frechet form if the limit of d[ln g(x)]/d(ln x) exists and is strictiy positive as x approaches infinity. Proof: We consider the von Mises convergence criterion of the Frechet domain of attraction for Y:

lim y fY(y)

y^ 1 - FY(y)

lim yfx[g~^(y)] dg-l(y)/dy

y^ l_Fx[g-l(y)]

lim g(x) fx(x) dx/dg(x) ~ x-^ 1 - Fx(x)

lim q(x) = k (13) X—><» X dg(x)/dx ^

^

If the limit of d[ln g(x)]/d(ln x) exists and is strictly

positive as x approaches infinity, from Eq. (4) the

distribution of the largest value from Y = g(X) is asymptotically of a Frechet form. 0

Theorem 6: The asymptotic distribution for the largest

value of the function Y = g(X) is of a Weibull form if there exists a finite O) such that co = sup{y : FY(y) < 1}

159
= k (14)

and the limit of [dx/dg(x)][a) - g(x)]/x exists and is strictly

positive as x ^proaches infinity. Proof: We consider the von Mises convergence criterion of the Weibull domain of attraction for Y:

lim (CO - y)fY(y)

y-^(0 1 - FY(y)

lim (0) - y)fx[g~^(y)]dg~^(y)/dy

y->0) l-Fx(g-l(y))

lim X fx(x) (0 - g(x) dx ~ X—>oo 1 - Fx(x) X dg(x)

lim co-g(x) dx x—>oo X dg(x)

If the limit of [dx/dg(x)][03 - g(x)]/x exists and is strictly

positive as x approaches infinity, then we conclude from

Eq. (5) that the distribution of the largest value from Y =

g(X) is asymptotically of the Weibull domain of

attraction. 0

Preservation and Transformation of the Weibull Domain of

Assume the random variable X to have an upper bound

w such that w = sup{x: Fx(x) < 1}, that the distribution of the largest value from X converges to a Weibull form,

and that X satisfies the von Mises convergence condition

in Eq. (5). Sufficient conditions are derived here under

which the asymptotic distribution of the largest value from

Y is preserved as Weibull or is transformed to Gumbel or FrecheL

Theorem 7: Assume Y = g(X) is unlimited in the

direction of the largest value. The asymptotic distribution for the largest value of the function Y = g(X) is of a Gumbel form if the limit of [(w - x)/g(x)] dg(x)/dx exists and is equal to zero as x approaches w and

[l/5jj(x)][dg(x)/dx]/g(x) is monotone for x close to w.

Proof: We consider the convergence criterion of die Gumbel domain of attraction in Eq. (3) for Y: lim 1 - FY(y)

y^oo yfY(y)

lim l-Fx[g-ky)]

y-^yfx[rky)]dg-i(y)/dy

lim 1 - Fx(x)

x^w g(x) fx(x) dx/dg(x)

lim 1 -Fx(x) (w - x) dg(x)/dx

X—>w (w - x) f(x) g(x)

lim (w - x) dg(x)

k x->w g(x) dx

(15)

If the limit of [(w - x)/g(x)] dg(x)/dx exists and is equal to zero as x approaches its upper bound w, then from Eq. (3) we conclude that the distribution of the largest value from

Y = g(X) is asymptotically of a Gumbel form if X

[l/5j^(x)][dg(x)/dx]/g(x) is monotone for x close to w. 0

Theorem 8: Assume that Y = g(X) is unlimited in the direction of the largest value. The asymptotic distribution for the largest value of the function Y = g(X) is of a Frechet form if the limit of g(x)/[(w - x) dg(x)/dx] is stricUy positive as x approaches its upper bound w.

Proof: We consider the von Mises convergence criterion of the Frechet domain of attraction for Y:

lim y fY(y)

1 - FY(y)

lim y fx[g~ky)] dg~^(y)/dy

y->°° i-Fx[g-i(y)]

lim g(x) fx(x) dx/dg(x) ~ X —> w 1 - Fx(x)

lim (w -x) fx(x) g(x)

= k

x -> w 1 - Fx(x) (w - x) dg(x)/dx

"im g(x)

(16) x-^w (w -x) dg(x)/dx

If the limit of g(x)/[(w - x) dg(x)/dx] exists and is strictiy

positive as x approaches its upper bound w, then from Eq.

(4) die distribution of the largest value from Y = g(X) is asymptotically of a Frechet form. 0

Theorem 9: The asymptotic distribution for the largest

value of the function Y = g(X) is of a Weibull form if there exists a finite co such that oo = sup{y : FY(y) < 1}

and the derivative dg(x)/dx exists as x apjproaches w.

Proof: We consider the von Mises convergence criterion of the Weibull domain of attraction for Y: lim (0) - y)fY(y)

y-XD 1 - FY(y)

lim (0) - y)fx(g~^(y))dg~ky)/dy

y-^G) l-Fx(g-ky))

lim (w - x) fx(x) CO - g(x) dx

= k

= k

x->w 1 - Fx(x) (w - x) dg(x)

lim co-g(x) dx

X—>w w - X dg(x)

lim dg(x) lim dx

x^w dx X—>wdg(x) = k (17)

where rH<^ital's rule is used in the second step fi^om the

last. It can be concluded from Eq. (5) that the ciistribution of the largest value from Y = g(X) is asymptotically of a Weibull form. 0

Summary of Conditions for Preservation and

Transformation of the Domain of Attraction

We emphasize here that the exact form of the monotonic transformation is not always needed to obtain

the domain of attraction of Fy. Consider, for example, the cases where Fx is any distribution that satisfies the von

160
Mises convergence criteria to be of a Frechet domain of

attraction, and g(X) is of the polynomial form, such that N

g(X) = p(X) = XaiXi (18)

i=0

and satisfying

N

d(^ix^)/dx>0 (19)

i=0

Since d[ln p(x)]/d(ln x) is equal to N as x approaches

infinity, the polynomial function p will always preserve

the domain of attraction of the underlying variable X, by

Theorem 5. Theorems 5, 6, 8, and 9 require only knowledge of the transformation g when the von Mises condition is assumed

to be satisfied by the underlying distribution Fx. The

X

remaining theorems require knowledge of both g and 5

^ where the second alternative (ii) of Theorem 1 , Theorems 4

and 7 assume the satisfaction of the von Mises condition

while the first alternative (i) of Theorem 1 and Theorems 2

and 3 do not necessarily assume satisfaction of the von

Mises condition for the underlying random variable X. For

engineering application, a table of the order of magnitude

X

of the function 5jj(x) for large x for some common

distributions is useful (Ref. [8]). ESTIMATION OF THE CONDITIONAL

EXPECTED VALUE

From the definition of the characteristic largest value of

Y,

n[l-FY(uJ)] = l (20)

Y X

u jj can be derived as a function dependmg only on u^

(Ref. [9]), nl =g(uj) (21)

From Eq. (7), 6^ can be expressed as a function of both

u^and5^:

sY=nfY(uY)

^gXMxl

n' dx .X

(22)

Y Y

Note that the formulas for u^ and 5^ do not rely on knowledge of the exact form of the underlying distribution Fx.

Reference [10] proposes a conditional expectation

E[Y I Y > F^^(a)] to represent the extreme risk realized

from the tail of the distribution, where Fy is the inverse

of the cumulative distribution function and a is a

partitioning probability that is chosen to bound from

below the range of extreme events. Building on this concept, Ref. [11] presents a result in approximating the

conditional expectation based only on the knowledge of the domain of attraction, the characteristic largest value Un

,

and the inverse measure of dispersion S^. The conditional

expectation E[Y= g(X) I Y > Fy (a)] can be obtained based on the identified domain of attraction of Fy without

knowledge of the exact form of the probability distribution of X. These approximations are nearly exact for large

values of n, and, equivalently, large values of the partition

probability a, since there is the relationship

n = -^

(23)

1 - a

between the selected partitioning probability a and the corresponding value of n (Ref. [9]). For a random variable Y of the Gumbel domain of

attraction, unlimited in the direction of the largest value,

and of an exfKDnendal tail (Ref. [2]), the approximation to E[Y I Y > Fy (a)] is given by (Ref. [11])

E[Y I Y > FY^(a)] = uX + -y" (24)

For a random variable Y of the Frechet domain of

attraction and of a polynomial tail (Ref. [2]), the approximation to E[Y I Y > F^\a)] is given by (Ref.

[11]) E[Y I Y > F^V)] = ul + ^+ (^)^(ul - ^7) (25)

For many (see the qualification in Ref. [2]) distributions of

the Weibull domain of attraction, the approximation to E[Y I Y > FY(a)] is given by (Ref. [11])

E[Y I Y > Fy^a)] = uX \

6^

[((o-uM + 1]

(26)

where co is the upper limit of Y. These approximations

are important in that no assumption of an exact

distribution Fy is needed.

EXAMPLE-FLOOD LOSSES

To illustrate the use of the derived results, we consider

the evaluation of extreme monetary flood loss when the underlying probability distribution of peak discharges is uncertain. Denote X to be the peak discharge in unit of

m^/sec and assume that statistical analysis of the peak

discharge record yields the following estimates of the

characteristic largest value and the inverse measure of

dispersion:= 6,000 m3/sec (27)

161
^= 1,500 m3/sec (28)

'100

and that the distribution Fx of peak discharges could be of

the Frechet domain of attraction. The stage-discharge

relationship is assumed to be of the following form for large flows (Ref. [12]): Y = g(X) = 3.92(y^f^« (m);

X> 1,000 m^/sec (29)

where Y is the stage in unit of m. The stage-damage

relationship h is assumed to be of the following form for high stages (Ref. [12]):

Z = h(Y) = 15,000,000 (l - ^-^^ ($);

Y > 1.10 (30)

where Z is the monetary flood loss in unit of dollar. Assume that the peak discharge Fx satisfies the von

Mises criterion to be of the Frechet domain of attraction. It can be concluded from Theorem 5 that the Frechet

asymptotic form is preserved in the stage Y since the stage-discharge function g is of the general form Y = X^,

where a > 0. Since the stage-damage function h is of the general form (o - cA' with c> 0, it can be concluded from

Theorem 6 that the Frechet asymptotic form in stage Y is transformed to a Weibull type asymptote in the loss Z. Although here the forms of the two monotonic functions

are given exactly, the domain of attraction of the resulting

functions Y = g(X) and Z = h(Y) could be obtained if they were known only by their general forms.

In two successive applications of Eqs. (21) and (22) the characteristic largest value and inverse measure of

dispersion of the stage Y and the loss Z are obtained as

follows: "100 = g("iaP

*100

"100

_ / 6,000 \0.30 ~^

V 10,0007 = 3.36304 m (31)

: X dx

>100 .1 X

"100

3.92

(1,500) n

(0.30)(6,000)-

(10,000)0-30

0.252228 m

-0.70

h(ui"^

(15,000,000) (

$10,093,666

1.10 3.363 )

(32)

(33)

J 1 dh(y)

>100 "100 = (0.252) (15.000,000)

1.10 (3.363)2 = $367,646.77 (34)

These results enable us to use Eq. (26) for Fz of the Weibull domain of attraction to calculate the conditional

expected value with a = 0.99 (and consequently n = 100)

as follows

E[Z I Z > F^^(0.99)]

1

^ + ^ _

§1^0

5i?o [(CO - ui?o)Sl?0 + 1] = 10093666 + 367647

367647

[(15000000- 10093666)7367647 + 1] = $10,435,684 (35)

The interpretation of this measure of extreme events is the expected flood loss conditional on either exceedance of the 99th percentile of loss or, equivalently, exceedance of the 100-year discharge. The conditional expected value of loss can be used in addition to the expected value of loss (and

other criteria) for selecting an optimal design (Refs. [13]

and [14]). CONCLUSIONS

This paper has studied the characteristics of extreme

realizations of a monotonic function of an underlying

random variable with an unknown distribution. The results make possible analyzing the extreme values of a univariate monotonic function with limited knowledge of the underlying distribution and of the exact form of the function itself. Estimation of the conditional expected

value of the system outcome, a measure of the risk of

extreme events, has been demonstrated when exact knowledge of the underlying distribution is unavailable.

The von Mises convergence criteria used as the basis for these results are sufficient conditions for determining the domain of attraction (Ang and Tang 1984). One future extension of this research is to derive rules governing the transformation and preservation of domains of attraction using conditions that are both necessary and sufficient ACKNOWLEDGMENTS

The authors are grateful to Professor Loren Pitt for assistance with the derivation in the Appendix and to Professor Enrique Castillo for consultation during the review of literature. This research was supported in part by

the National Science Foundation under grant No. BCS8912630.

APPENDIX. EQUIVALENCE OF TWO

CONVERGENCE CRITERIA FOR GUMBEL

DOMAIN OF ATTRACTION

Assume that X is unlimited in the direction of the

largest value and define

IT/ ^

1 - Fx(x)

"^"^ = fx(x)

where Fx(x) and fx(x) are the cumulative distribution function and the probability density function of X,

respectively.

Theorem Al: If H(x) is continuous, differentiable, and

monotone for large x, then

lim H(x)

X—>«« = 0 (37)

162
implies that lim d

H(x) = 0 x-^ dx

Proof: Note that H(x) is always nonnegative. We have

H(2x)

2x

(38)

= H(x)+jH'©d4

2x

(39)

It follows from Eq. (39) and the mean value theorem that there exists some y e [x^x] such that H(2x) > (2x - X) H'(y) (40) In the limit as x approaches infinity, dividing Eq. (40) by

2x yields

lim H(2x) ^ lim H'(y)

x-^ 2x X—)oo 2

y e [x,2x] (41)

If H'(x) > 0 for large x, then Eq. (38) follows from Eqs.

(37) and (41).

It remains to consider the case of H'(x) < 0 for large x. Rearranging Eq. (39), we have

H(x)

2x

H(2x)- Jir(^)d^

2x

(42)

It follows from Eq. (42) and the mean value theorem that there exists some y e [x,2x] such that H(x) > -<2x - x) H'(y) (43)

In the limit as x approaches infinity, dividing Eq. (43) by

X yields

lim H(x) lim > -H'(y) y G [x,2x] (44)

If H'(x) < 0 for large x, then Eqs. (37) and (44) lead to Eq.

(38). 0

Theorem A2: If H(x) is continuous and differentiable

for large x, then

fH(x) = 0 (45) x^oo dx

implies that lim H(x)

X—><» X

Proof: If Eq. (45) holds, then for any e > 0, there exists a number N such that tH'(x) I < e for x > N. We have from

the triangle inequahty

H(x}

= 0 (46)

= [H(N)

< |H(N) I /x+ I Jh'(0 d4 I /X

< IH(N) I /x + e (x - N)/x (47)

Taking the limit of Eq. (47) as x approaches infinity gives 'TM 55i5i<e (48) X->oo X

Since e can be chosen arbitrarily small (N arbitrarily

large), the Eq. (46) follows immediately from Eq. (48). 0

REFERENCES

[I] Gumbel, E.J., Statistics of Extremes, Columbia

University Press, New York, 1958.

[2] Ang, A. H-S. and W. H. Tang, Probability Concepts

in Engineering Planning and Design, Volume II: Decision,

Risk, and Reliability, John Wiley and Sons, New York,

1984.

[3] Castillo, E., Extreme Value Theory in Engineering,

Academic Press, Boston, 1988.

[4] Johnson, N.L. and S. Kotz, Continuous Univariate

Distributions, John Wiley and Sons, New York, 1970.

[5] Galambos, J., The Asymptotic Theory of Extreme

Order Statistics, 2nd edition, John Wiley and Sons, New

York, 1987.

[6] de Haan, L., On Regular Variation and its Application

to the Weak Convergence of Sample Extremes,

Mathematical Centre Tracts, Amsterdam, Vol. 32, 1970.

[7] von Mises, R. von. La distribution de las plus grande

de n valeurs, Mathematique, Interbalkanique. 1 (1936),

141-160.

[8] Lambert, J.H. and D. Li, Evaluating risk of extreme

events: results for monotonic functions. To apf>ear in Journal of Water Resources Planning and Management,

May 1994.

[9] Karlsson, P-0. and Y.Y. Haimes, Risk-based analysis

of extreme events. Water Resources Research, 24 (1988),

9-20.

[10] Asbeck, E.L. and Y.Y. Haimes, The partitioned

multiobjective risk method. Large Scale Systems, 6

(1984), 13-38.

[II] Mitsiopoulis, J.A., Y.Y. Haimes, and D. Li,

Approximating catastrophic risk through statistics of

extremes. Water Resources Research, 27 (1991), 1223-

1230.

[12] Haimes, Y.Y., D. Li, and V. Tulsiani. Integration of

Structural Measures and Flood Warning Systems for Flood

163
Damage Reduction, Report to The Institute of Water

Resources, U.S. Army Corps of Engineers, Fort Belvoir,

Virginia, Prepared by Environmental Systems Modeling,

Inc., Charlottesville, Virginia, 1992.

[13] Karlsson, P-0. and Y.Y. Haimes, Risk assessment of

extreme events: An application. Journal of Water

Resources Planning and Management, 115 (1989), 299-

320.

[14] Haimes, Y.Y., J.H. Lambert, ?.nd D. Li, Risk of

extreme events in a multiobjective framework. Water

Resources Bulletin. 28 (1992), 201-209.

164
Second Order Behavior Of Domains Of

Attraction And The Bias Of Generalized

Pickands' Estimator

Pereira, T.T.

University of Lisbon, Lisbon, Portugal

The domain of attraction of the generalized extreme value distribution is studied with respect to second

order conditions using the tail quantile function. The particularly important case of the differentiable domain of attraction is emphasized. Next the weak consistency of a generalization of Pickands' estimator

for the main parameter of an extreme value distribution is proved. Moreover, under quite general conditions on the underlying distribution function, that include second order behaviour and being in the differentiable domain of attraction, the asymptotic normality of the estimator is proved and the asymptotic bias that can

occur is determined. A result concerning the minimization of the asymptotic mean squared error of the estimator is given which leads to an optimal choice of the number of intermediate upper order statistics involved in the definition of the estimator. Several examples, including all the usual continuous

distributions, illustrate the results. Suggestions on how to choose the parameters in practical applications

are made.

1. Introduction

The classical extreme value theory is primarily concerned

with the asymptotic distribution of the maximum of

independent and identically distributed (i.i.d.) random

variables. Let Xj, i>l, be a sequence of i.i.d. random

variables with distribution function F and let Xn;n=max(Xi,...,Xn), n>l. Gnedenko, Ref. [1], proved

that, if there exist sequences of real constants an and bn,

n>l, with aii>0 and a nondegenerate distribution function

G such that

lim P((Xo:n-bn)/an ^) = lim F°(anX+bn) = G(x) (1.1) n— n—

for all X at which G(x) is continuous, then G(x) belongs

to one of the three types of extreme value distribution

functions. Using a parametrization of von Mis6s, Ref. [2], these limiting types can be written in a unified way,

known as Generalized Extreme Value (GEV) distribution,

Gy(x) = exp{-(l+7x)-i^T^}, l+yx >0, ye E

.

We say that F belongs to the domain of attraction of Gy,

notation FeDCGy), if (1.1) holds for some sequences an and bn. The characterization of the domains of attraction of extreme value distributions has been dealt with by several

authors. Von Mis6s, Ref. [2], found sufficient conditions

for a distribution function to be in die domain of attraction of each of the three extreme value distributions and

Gnedenko, Ref. [1], gave necessary and sufficient

conditions. More recently, de Haan, Ref [3], and

Pickands, Ref. [4], presented a unified characterization of

the domain of attraction of the GEV distribution. Let the function U be defined by U(x) = (1/(1-F))^(x), x>l,

where the arrow denotes the (generalized) inverse function. Note that U(x) = Q(l/x) with Q the quantile function of

the upper tail of F. A necessary and sufficient condition

for a distribution function F to be in the domain of

attraction of Gy, for some ye E , is the existence of a

positive function a(.) such that, for x>0.

165
lim^M^l.SM (,eadlogx,ifr=0) (1.2) t—><» dyl) Y

(Ref. [3]). Moreover the auxiliar function a(.) is regularly

varying with index y, i.e., aeRVy (cf. Ref. [5], th.1.9.).

Pickands, Ref. [4], considered the inverse of the hazard

cumulative function, H"! (x) = (1/(1-F))^(e'^) = Vie^), to

give a characterization of the domain of attraction which is

easily seen, through a logarithmic transformation, to be

equivalent to the one of de Haan. The necessary and

sufficient condition (1.2) for the domain of attraction of Gy

can of course be written

U(tx)-U(t) _

xV-1

a(t) y + R^,x(t) with Ry,x(0 = 0(1), t-^oo.

(1.3)

If there exists a positive function R(t) such that R(t)=o(l),

t-^oo, and RY,x(t) = hY(x)R(t) + o(R(t)), we can speak of

second order behavioiu' of U. We then have for x>0.

U(tx>-U(t)-a(t)-

xY-1

lim

t—>oo a(t)R(t) = hy{x) (1.4)

and our purpose in section 3 is to find the possible limit functions hy(x) with hy(x) finite and not constant. Besides we will find the function R(t) which describes the rate of

convergence of the limit (1.4) to be a regularly varying

function with index p for some p<0. We also analyse the second order behavioiu' of U for a distribution function in the differentiable domain of attraction of Gy, and we

identify the generalized Pareto distribution as the only

family of distribution functions with positive derivative in this subdomain of attraction for which Ry,x(0 is

identically zero. Related papers are Ref. [6], Ref. [7] and

Ref. [8]. Section 4 deals with the problem of estimafing the extreme value index y from a finite sample Xi,X2,...,Xn.

A semi -parametric approach is due to Pickands, Ref. [9]; he proposed the estimator

aP

2(n)_ ^(n)

m 2m

y; = (log2)-ilog —

'2m

,(n)

Mm

l<m<[n/4]

where Z^j°^>Z2°^>...>Zp"^ are the descending order

stafistics of Xi,X2,...,Xn and m=m(n) is an intermediate

sequence of integers, i.e., m->oo and m/n^O (n^«>).

Pickands proved that his estimator (based on the 4m

largest observations) is weakly consistent and Dekkers and

de Haan (1989) proved this same result and showed that if the sequence m(n) increases suitably rapidly the estimator

is also strong consistent. Moreover, tmder additional

conditions on the distribution (differenfiability and 11- variation of ±t^"'iU'(t) for real y or second order regular

variafion conditions on U for y^K)), they proved the asymptotic normality of the estimator for intermediate

sequences m(n) which increase at certain rates. Note that with a reparametrization the estimator can be written

-(n) _ (n)

^[m/4] '^[m/2]

'^[m/2]

based on the m largest observafions. We consider the following generalization of Pickands' estimator

yn.e = Hoge)-Uog-

(n) _ (n)

[mQ^] [me]

^(n) _

(n)

"^[me]

, o<e<i,

where we adopt the convention

r 1 / 1 ' 0<X<1 ^, ^ ap aP

t^]= \ largest integer<x, x>l • Note that y„=y„,i/2-

This estimator is weakly consistent for any 0 in the

interval ]0,1[ and for any intermediate sequence m(n).

Under quite general conditions on F (F in the differentiable domain of attraction of Gy and second order behaviour of

aP

U), we shall prove the asymptotic normality of yn,e for ^

certain rate of growth of the intermediate sequence. These

conditions are more general than the ones considered in Ref. [7]; namely, tiiey include the case of second order

regular variation behaviour for y^O which was not taken

into account in Ref. [7]. Also our proofs do not use the argiunents in Ref. [7] but are based on the asymptotic

joint distribution of a fixed number of intermediate order

statistics associated to the same intermediate sequence

established by Cooil, Ref. [10] and Ref. [11]. Moreover we shall give the asymptotic bias of the estimator that occurs if the sequence m(n) is allowed to increase at a

166
faster rate, as well as a theoretical result concerning the minimization of the asymptotic mean squared error of

generalized Pickands' estimator. For a review of the different approaches (parametric and

semi-parametric) to the estimation of the tail of a

distribution see Ref. [12]. In section 5 we will ilustrate the results with the

continuous distribution functions that are typically used in

statistical applications.

Finally, in section 6 we make some considerations on how to choose die parameters 9 and m in pratical

applications, and we suggest the use of the particular value

of 9 (corresponding to one of the possible generalizations

of Pickands' estimator) which minimizes the variance of

the estimator when ^=0.

2. Preliminary results

The importance of regular variation theory in the

characterization of domains of attraction is shown

in the next theorem, where U(<») = lim U(t) and

t—>oo

Xco = sup{x: F(x)<l}.

THEOREM 2.1.(Gnedenko, Ref. [1]; de Haan, Ref. [3]) (1) For Y>0 are equivalent: (i) FeD(GY), (ii) U(c«)=oo

and Ue RV^, (iii) x^=°° and 1-Fe RW-uy.

(2) For Y<0 are equivalent: (i) Fe D(Gy), (ii) U(~)<oo and

U(oo>-U(x)e RVy, (iii) Xoo<~ and l-F(Xoo-x-i )e RWyy.

(3) For Y=0 are equivalent: (i)F6D(Go), (ii) there

exists a positive function f such that for real x.

lim

l-F(t+xf(t))

l-F(t) = e-^ (i.e., 1/(1-F)er(f)), (iii) Uen(a).

WhenF6D(GY) ^ ^ ^^^^ ^^^^^

regtilar variation tail behaviour and that U has a first order

regular variation behaviour. When FgD(Go), we say that U has a first order Il-variation behaviour; moreover, the

functions a(.) and f(.) are related by a(t) = f(U(t)). The differentiable domains of attraction were introduced by

Pickands, Ref. [4]. We say that F belongs to the (once)

differentiable domain of attraction of Gy, notation

F^DdiKGy). if the distribution function F is differentiable

in a left neighborhood of x<„ and if there exist sequences an>0 and bn such that

lim [F°(anx+bn)] = iGy)(x) (2.1) n—»«« ux '

uniformly for all x in any finite interval. Clearly,

FeDdif(Gy) implies F€D(Gy) for the same attraction coefficients an>0 and bn. Condition (2.1) can also be

written

lim nanF(anX+bn) = (l+7x)"^^Y-i n—>»

and, in particular for x=0 we have lim nanF (bn) = 1, that n—»<»

is, an ~ l/(nF(bn)). If F is positive and we take bn=U(n)

this allows us to consider an = nU'(n). The following

theorem characterizes the differentiable domain of

attraction of Gy.

THEOREM 2.2. (Pickands, Ref. [4]) A distribution function FeDdiiKGy) for some ye R if and only if H~i (t)

is differentiable for all sufficiendy large t and, for real x.

(H-i)(t+x) hm —,— (H-i)(t) = eYx . (2.2)

It is easily verified that condition (2.2) is equivalent to (H"i ) (logt) being a y-regularly varying function and, as H-i(t) = U(et), (2.2) can be written tUXDeRV^. Hence,

the above dieorem can be restated in terms of the tail quantile function U.

COROLLARY 2.1. FeDdiKGy) for some ye E if and

only if U(t) is differentiable for all sufficiently large t and

ti^'(t)eRVo.

Cooil, Ref. [10] and [11], proved that if FeDdiKGy) for some ye E then for any intermediate sequence m=m(n)

and 9>0, the stochastic process

a)L°\e)

^n/m

where bn/m9 = F^(l-m9/n) = U(n/m9), converges in

distribution, n— to the gaussian stochastic process

(0(9) characterized by

E((o(9)) = 0, 9>0,

Cov(co(9i),co(92)) = 9i-^92-^-i , 0<9i<92.

From now on we shall consider the normalizing sequence

bn = U(n).

167
for sufficiently large t

3. Second order conditions for domains of

attraction

The next theorem identifies the generalized Pareto

distribution, Ref [9], as the only family of distribution functions belonging to DdifCGy) and having a positive

derivative for which Ry,x(t) can be identically zero. THEOREM 3.1. Let FeDdifCGy) for some ye E and

suppose F has a positive derivative F . Then are

equivalent:

(i) there exists a positive function a(t) such that, for x>0,

U(tx)-U(t) _

xV-1

a(t) Y

(i.e., RY,x(t) = 0).

(ii) F is the generalized Pareto distribution function

F(x)= I 1+Y~^ j

. x>C2, l+Y-^>0, ci>0, C2eE.

PROOF. The existence of a positive derivative U of U

allows us to say that (i) is equivalent to (i') "there exists a positive function a(t) such that tU'(tx)/a(t) = y^~^ for all x>0" and taking x=l in (i') we obtain a(t) = tU'(t).So, the functional equation to be solved is U'(tx)AJ'(t)) = x^^"^ for x>0 and t>max{l,l/x}. For x>l we can take in particular t=l which leads to U'(x) = x'y~'U'(l), x>l, which is equivalent to U(x) = C](xT^-1)/y+C2 for some Ci>0, C26 E.

For 0<x<l we can take in particular t=l/x and get

U'(l/x) = U'(l)xi-l', 0<x<l, which is equivalent to U'(x) = xV-iU'(l), x>l.

Note that for any odier function a(t) asymptotically equal

to tU'(t) the generahzed Pareto distribution verifies (1.3)

with Ry_x(O*0. What the theorem implies is that Ry^(t)^0 for any other distribution function FeD(jif(GY)

with positive derivative, whatever the possible function

a(t) considered. As can be seen in the proof of the theorem,

(i) is also equivalent to t'"^'(t) = ci, ci>0.

LEMMA 3.1. Let Fg D(Gy) for some ye E . If there

exists a positive function b(t) such that, for x>0,

„,

«x)-ra(»)-,-Va(.) ^^^^^ (3.1,

t-><»

with h^(x) finite and not constant, then

* xP-1

h.y(x) = c , x>0, for some c-^, p<0, (3.2) P

the case p<0 being possible only if linit-Ya(t) = d>0.

Moreover b(t)e RVp and b(t) = o(t-Ya(t)).

PROOF. Remember that FeD(Gy) implies t-Ya(t)GRVo.

The finite and not constant limit function of

[(tx)-Ya(tx) - t-T^a(t)]/b(t) is c(xP-l)/p for some pe E and

(cf th. 1.9, Ref.[5]). But p>0 implies t-Ya(t)eRVp

(cf th. LIO, Ref[5]) which contradicts the fact of

FeD(GY). If p<0, then lin^t~Ya(t) exists and moreover

+(( lim t-Ya(t)) - t-Ya(t))eRVD, with the plus sign if oO t

^ and the minus sign if c<0. Now, if t"1^a(t) = 0 we will t—>oo have t~Ya(t)e RVp with p<0 and, again, this is not

possible; if limt-Ya(t) = d>0, then ±(d-t-Mt))e RVq

which implies that t~'^a(t)e RVq and only this last

situation is possible if p<0 and for a distribution function

F in the conditions of the theorem.

Note that if (3.2) holds with p=0, ±t"T'a(t) belongs to the

class n ( ±t-Ya(t)€ n(ai) with ai(t) = lclb(t) ) and if it holds widn p<0, t~'^a(t) is a slowly varying function but

not a Il-varying one and ±(d-t~'i^a(t))e RVp. A particularly

important case of the above lemma is the one of

FeDdiKGy) with U having a positive derivative U' and a(t) = tU'(t): if p=0 we will have ±ti^'(t)en and

if p<0 then jii^t^~YU'(t) exists and is positive and

±((tlilS=t^"^'(t)) - ti-nj'(t))e RVp. The following

theorem shows that U will then have a second order

behaviour and idendfies the function R(t) and the possible

limiting functions hY(x).

THEOREM 3.2. Let FeDdiKGy) for some ye E and

suppose that U admits a positive derivative U'. If (3.1)

holds for a(t) = tU'(t), then for x>0.

U(tx)-U(tHU'(t)^

lim —=hy(x)

t-»oo tYb(t)

2 log^x

withhY(x)=^ ^

[xYlogx

c FxY^ -1 _

x_Y-l

p L y+p y J

for some ct^O.

y=0 p=0

y^O p=0

ye E p<0

168
PROOF. For x>0. PROOF. For x,y>0.

U(tx>-U(tHU-(t)^ f(t3)i^(tsH^-^(t) '— , 1, = — sY"Ms.

frhit) J b(t)

U(txy)-U(t)-a(t)

a(t)R(t)

(xyF-1

If p=0 this integral converges to c js^ Mogsds, t^oo

(th. 1.14., Ref. [5]). If p<0 we easily see that ^lim(ti^(t) - d)/b(t) = c/p and as the above integral is X

ti-rU'(t) - d f

/(ts)i-YUXts) - d ^ , _i , equal to — |

(

r~r: 1 m Ms it b(t) ti-nj'(t)-d

converges to - j(sP-l)sT^ids, t-^c>o (th. 1.3., Ref. [5]).

For p=Y=0 the converse statement is also true (cf. Ref.

[6]). In a wider context a related result concerning the possible

limiting functions can be obtained if we assume a second

order behaviour for U. Again the important limit will be

the one of (3.1). A similar result was obtained in Ref. [8] for the function logU.

U(txy)-U(tx)-a(tx)

yY-l

Y a(tx)R(tx)

a(tx)R(tx) a(t)R(t)

U(tx)-U(t)-a(t)

a(t)R(t)

xY-1

V (tx)-ya(txH-ya(t)

-<—+ xY Y t-T^a(t)R(t)

(3.4)

(tx)^a(tx)-t-T^a(t)

(a) If the limit of , as t->«>, exists for all t-^a(t)R(t) x>0 and the limit function is not constant, it will be of

the form Ci(xP-l)/p, for some ci^^O and p<0 (p<0 only if lim t-Va(t) = d>0, t-»oo) and t-'Va(t)R(t) is a p-regularly

varying function (Lemma 3.1). Hence R(t)£RVp. Taking

limits, as t^°°^ in (3.4) one obtains the following

functional equation

h^Cxy) = hy(y)xY+P + hy(x) + CixY

yY-lxP-1

(3.5)

THEOREM 3.3. Let FeD(GY) for some ye E and

suppose that (1.4) holds with hy(x) finite and not constant.

If, in adittion, one of the following conditions holds

(a) lim

(tx)-Ya(tx) - t-Ya(t)

t^a(t)R(t)

limit function is not constant,

(tx)-Ya(tx) - t-Ya(t)

exists for all x>0 and the

(b) t—> » t-^a(t)R(t) = 0, R is measurable and

lim R(txi

exists and is finite for all x>0,

R(t) then hy(x) belongs to one of the following classes

^ log^x Cr^—+ C2lOgX,

Ci/^xP-l \ xP-1 ^—-logxUc^-

yl^xYlogX-—jf C2—

Cifxf>^y-l xT^-n XP+Y-1

*<pl p+Y y J p+Y

with ci?0 and C2G E or ci=0 and C2?0.

Y=0, p=0, (1)

Y=0, p<0, (2)

Y^^O, p=0, (3)

y^O, p<0, (4)

(3.3)

Assuming that hY(x) is a differentiate function, it

follows by differentiation with respect to y that

xP-1

xh^(xy) = h^(y)xY+P + cjxY ^

yY"i and setting y=l

yields h^(x) = h^(l)xV+f>-i + cixY-i(xP-l)/p which, noting

that hY(l)=0 and h^(l)e E, leads to

log^ , ci—^+C2lOgX

h^(x)=<

ci yp-i ci P ^ P P

^

Y=0, p=0

Y=0, p<0

/Ci \XP+Y-1 ci xY-1

with Ci?tO and C2e E . Let h^(x) be the difference between

the general solution of the equation (3.5) and the

differentiable one, i.e., h!^(x) = hY(x) - li^(x). Then h!^x)

169
satisfies the equation h^(xy) = hi^(y)xP+'y + h!^(x). Also by

symmetry h!^(xy) = h!^(x)y'>'+P + h^(y) and subtracting we

get h^y)(xY+P-l) = h^(x)(yV+P-l) for x,y>0. Hence

h^(x)/(xT-*-P-l) is constant, i.e., h!J,(x) = c(xP+Y-i)/(p+Y),

07^0 for all x>0 (read clogx for p+y=0) or h^(x)=0 (because

h^(x) is a solution of (3.5)). Then h^(x) = c(xP+V-i)/(p+Y),

ce E, and hY(x)=h^(x).

. r

(tx)-ya(tx) - t-Va(t) ^ . „

(b) Suppose now that lim = 0 for all t->oo t-^a(t)R(t) R(tx) x>0 and let f(x) = lim ^, , which exists and is non

t^oo R(t)

negative for all x>0. Since ^^tt^ = tBi^ we have ° R(t) R(tx) R(t)

f(xy) = f(y)f(x) for all x,y>0. As R is measurable the

function f(x) is also measurable and the only measurable

solutions of Cauchy functional equation are f(x)=xP for some pe E and f(x)=0 for x>0. However this last solution

is not compatible with f(l)=l. Hence f(x)=xP, for x>0, for some pe E and R(t)€ RVp. But as R(t)=o(l), t->oo, it has

to be p<0. This leads to the functional equation

hy(xy)=hY(y)xY+P+by(x). Hence, hy(x) = c(xP+Y-i)/(p+Y),

ctK) for all x>0 (read clogx for p+Y=0). Note that these are

the classes of (3.3) with ci=0 and C29^.

In both theorem 3.2 and theorem 3.3 we have for the function R(t) describing the rate of convergence of the

limit (1.4) that R(t)eRVp for some p<0. If U has a second

order behaviour and the limit (3.1) exists finite and not

constant with p<0 (p=0) in (3.2) we say that U has a second order regular variation (Il-variation) behaviour.

4. Asymptotic normality and bias of

generalized Pickands' estimator

aP

THEOREM 4.1. (Weak consistency of Yn,e)

If FeDdif(Gy) for some ye E, m=m(n)->oo and m=o(n),

aP p n—>oo, then Yn-G n-><». p

( stands for convergence in probability)

PROOF. The asymptotic distribution of z[^\^- Z^°\

0<e<l (Ref. [10]),

(n) (n) ^[me]~^m -(''n/me-Dn/m)

W

n-^, implies that

-4 N(O,l-2e-Y+0-2Y-i),

(Z|^e]"^m^^/^n/m = (bn/me-bnymVan/m + Op(l/Vm)

= (0-Y-l)/Y+Op(l),.

i.e., (Z|^e]-^m^)/^n/m ^ O-^-D/y, n-^oo. (4.1)

Hence, for O<0<1,

Am,e - ^^[nj02] ^[mefnme] ^-

= - zL°V(Z;°^«, - Z^"V 1 ^ G-Y

aP P

and Yn.0 = logAin,e/(-loge) log0-Y/(-loge) = YHowever, the weak consistency of Yn.e for any

intermediate sequence m(n), as well as the strong

consistency for an intermediate sequence m(n) such that m(n)/loglogn^oo, can be proved for F in D(Gy) by using

the argument in Ref. [7].

aP THEOREM 4.2. (Asymptotic normality of Yn.e) Let Fg DdiKGy) for some Y^ ^ . m=m(n) be an

intermediate sequence (m=m(n)— and m=o(n), n^«>) and

O<0<1.

(A) Suppose that for some normalizing function a(.),

U(t/0) - U(t) e-y-1 . n M

a(t) " ~7~' i-^-' RY.e-i(t)-0- (4.2)

/— aP

Then Nm(Yn.e-Y) has asymptotically a normal

distribution with mean value zero and variance a^{y,Q) for any intermediate sequence m(n).

(B) Being Ry,e-i(t) not identically zero, suppose that U

has a second order behaviour, that is, there exists a

positive function R such that.

170
Ry,e-i(t) = by(e-i)R(t) + o(R(t)) and R(t)=o(l), (4.3)

Then\m(yn,e - Y)

(i) has asymptotically a normal distributioii with mean

value zero and variance a^(Y,6) for intermediate sequences

satisfying m=o(nyg^(n)),

(ii) has asymptotically a normal distribution with mean

value bc(Y,6) and variance o^(y,6) for intermediate

sequences satisfying m - n/(g^(n/c^)), oO,

where g(t)=t/R^(t) and g^ is the asymptotic inverse

ftinction of g,

bc(Y,e)=cb(Y,e)=c

cu. ^^^^^— (bc(o,e)=c^) / 9^(0-^-1 )(-ioge) log^e

and H(y,e) = hy(r2H0"^+l)hy(e-i ).

REMARK. In view of Corollary 2.1, Lemma 3.1 and

Theorem 3.2 the conditions of Theorem 4.2 include the following ones (with ye E ),

(A')..^(,).RVoandlMl=5^.

(B')ti^(t)eRVo and for x>0,

lim

(tx)^-^'(tx) - ti-yU'(t) _

xP-1

t-^oo b(t) ~

p

for some c;^, p<0 and some positive function b(.).

(B") ti-^(t)e RVo and ±ti-nj'(t)en or ±(d-ti-Taj'(t))e RVp with d = jim ti-YU'(t)>0 for some

t—

Nm,e = Vni-

^[me] ~ ^m^ ~ (''n/me " bn/m)

^nJm

„(n) „(n)

^[me2] ~ ^ [mO] ~ (bn/me2 - bn/mO)

^n/m

Nm^e2 = Vmand note that Tm^e has asymptotically a normal

distribution with mean value zero and variance

y2(e-'-i)(i+e-2i^i)e-2Y

Va = ~

. The right hand side of

(4.4) is equal to

y J

bn/ine2-t>n/me-Q~'^(bn/me-bn/m) Tm9+——vm (4.5)

If the second term is zero or if we are able to make the second term negligible by letting the sequence m(n)

increase appropriately, the asymptotic distribution of

Vm(Ain,0 - 6"^ will be the one of Tjji 0. If the sequence

converges to infinity in such a way that the second term converges to a constant different from zero, then the asymptotic distribution will still be normal with the same

variance but with mean value different fi"om zero. Case (A) - If the distribution function F is such that for some sequence of attraction coefficients an we have

bn/me-bn/m U(n/me)-U(n/m) 6-^-1

^n/m an/m

, n>l, then

p<0.

Vmbn/me2 - bn/i

an/m

(B-y+iy

bn/m9 - bn/m

an/m = 0, n>l

PR(X)F. We begin by investigating in what conditions

'^(Am,e - 6""^) bas asymptotically a normal

distribution. By (4.1)

Vm(Am,e-e-^ ~ Vm

(An) in) (n) (n) 1

an/m(e-T^-i)/y

and ^ m( Ain,e - 9~'>') has aymptotically a normal

distribution with mean value zero and variance for any

intermediate sequence m(n). Note that for any other

sequence of attraction coefficients an*, we have

an*/an=l+o(l) and the second term is also zero. Case (B) - If Ry 9-i(n/m) is not identically zero for any

possible choice of the normalizing sequence an we obtain

in probability (n->o=)

Now we introduce for 0<9<1,

(4.4) bn/me2-bn/m bn/me-bn/m

an/m

171
= Ry,e-2(n/mHe"^+l)RY,9-i(n/m)

= [hy(0-2)-(0-Y+l)hy(e-i)]R(n/m) + o(R(n/m))

= H(y,e)R(n/m) + o(R(n/m)).

Hence the second term of (4.5) is equal to

H(Y,e)VmR(n/m) + Vmo(R(n/m)). (4.6) 6^—1

If the intermediate sequence m(n) is such that Vni R(n/m) -^1, n-^oo, (4.7) which is equivalent to n ~ g(n/m), with g(t) = t/R^(l),

which in turn is equivalent to m ~ n/g^(n), n->oo, (4.8)

with the asymptotic inverse function of g, the second

term of (4.5) converges to bA(Y,e) = YH(Y,e)/(e-'i'-l) and

m,6 - 9 ^ has asymptotically a normal distribution with mean bA(Y'6) variance Va- Note that g(t)-»oo,

Let mo(n) be a sequence satisfying the above condition,

that is, mo ~ n/g^(n), (4.9)

and let m(n) be any other intermediate sequence of smaller

order than mo, m=o(mo), n—>oo. (4.10)

We are going to show that for the sequence m(n)

satisfying (4.10) the second term converges to zero. In

fact, we know that for the sequence mo we have

V moR(n/mo) —>1, n— Now

V^(n/m,.^/^R(„/,„0)TM

Noting that m=o(mo) is equivalent to n/mo=o(n/m) and

that R(t) —>0, t^oo, we conclude that R(n/m) can not be

of a greater order than R(n/mo) and hence R(n/m)/R(n/mo)

can not converge to infinity. Then VmR(n/m) —>0, n—

Hence, for any intermediate sequence

I— w

m=o(mo)=o(n/g^(n)), Vm(Am,e-e-V) -> N(0,Va), n^oo

(case(BHi)).

Assume now that the intermediate sequence m=m(n) is such that VmR(n/m) c, n^oo, c>0, (4.11)

that is, n ~ c^g(n/m), n-»cc, which is equivalent to

m -

-(n/c2)

, n—>oo. (4.12)

In this case Vmi(Ani,e-9 "0 has asymptotically a normal

distribution with mean cbA(Y.0) and varianceVa (case (B)-

(ii)). aP

Now to obtain the asymptotic distribution of Yn-e we just

have to expand it about 9^ since the estimator is equal to logAm,e/(-log9). From

aP Am,e-9-T

yB,Q=y + + o(Am,0-9-TO

we obtain

V^CYn.e - Y) = ^£St^+ o(V^(Am,e-9^)

9-^(-log9)

An, fl-9-Y+Op(l)

9^(-log9)

Am,e-9-T^

9-l^(-log9)

^r- Am,G-9-Y = \ m

-0^(-log9)

i— aP

and finally we conclude that vm(Yn,e-Y) has an

asymptotic distribution which is,

in case (A), N(0,VA/9-271og2e) = N(0,a2(Y,9))

for any sequence m=m(n)—>«> and m=o(n).

in case (B)-(i), N(0,Va/9-2Ylog^G) ^ n(0,o2(y,9))

for sequences m=m(n)^oo and m=o(n/g*~(n)).

in case (B)-(ii), N(cbA(Y,9)/9-^(-log0),VA/9271og2e) = = N(bc(Y,9),a2(Y,9))

for sequences m=m(n)—>oo and m - n/g^(n/c2).

The first question which naturally arises is the one of

knowing what are the distribution functions belonging to the differentiable domain of attraction of the GEV

distribution for which the asymptotic normality of the estimator is valid for any intermediate sequence. Corollary

4.1 gives an answer to this question showing that from

among the distribution functions belonging to the

differentiable domain of attraction of GEV distribution which admit a positive derivative only the generalized

Pareto distribution verifies case (A) of theorem 2.2. COROLLARY 4.1. Let Fe Ddif(Gy) for some ye E

and suppose that F admits a positive derivative F. Then

/— aP

\m(Yn,e~Y) has asymptotically a normal distribution

172
with mean value zero and variance a^(Y,9) for any

intermediate sequence m(n) if and only if F is the

generalized Pareto distribution function.

PROOF. Note that for a distribution function F in the r- aP

conditions of the theorem vm(Yn,e-Y) is asymptotically

N(0,a^(Y,Q)) for any intermediate sequence m(n) if and

only if U(n/me2)-U(n/m)-(e-Y+l)(U(n/me)-U(n/m)) = 0, (4.13)

as is easily seen from the proof of theorem 4.2. We are going to solve the functional equation

U(tx2)-U(t) = (xT^+l)(U(tx)-U(t)) for t>l and x>l. (4.14)

tx This equation is equivalent to JU'(y)dy = xYjU'(y)dy or

tx t tx tx JU'(xz)xdz = x'i^JU'(z)dz which in turn is equivalent to

t t U'(xz)-xT^^U'(z)=0. In particular for z=l, this equation

reads U'(x) = xY-^U'(l) which is equivalent to U(x) = ci(x'1^-1)/y+ C2, Ci>0, cie E (= cilogx+C2, Y=0) which

is the tail quantile function of the generalized Pareto

distribution. Moreover equation (4.14) is equivalent to (U(tx)-U(t))/tU'(t) = (xY-l)/Y which means that the normalizing sequence an such that (4.2) holds is an=nir(n).

If we assume R(t) to be a regularly varying function

(which appears to be a natural restriction in view of

previous section results) the conditions on the sequence

m(n) stated in the above theorem admit a slight

simplification and furthermore it is possible in most cases

to minimize the mean squared error of the estimator when

the bias is different from zero. COROLLARY 4.2. If the function R(t) of theorem 4.2

is p-regularly varying at infinity then p<0 and g^ is also

a regularly varying function at infinity with index

p*=(l-2p)"^ (0<p*<l). Moreover condition (4.12) of

theorem 4.2 is equivalent to

m - c^P

* g^(n)

oO. (4.15)

PROOF. Let R(t)eRVD. As liniR(t)=0 it has to be p<0

(cf. corollary 1.2.1, property 1, Ref. [13]). This is equivalent to saying that g(t) = t/R^(t) is regularly varying

at infinity with index l-2p, with l-2p>l. On the other

hand, there exists a function V(t)eRVi_2p and sfrictly

monotone which is asymptotically equal to g(t) (cf. corollary 1.2.1, property 7, Ref. [13]). It follows from

g(t)-^c«, t—>oo, that V(oo)=oo and hence V is a strictly

increasing function. This implies (cf. corollary 1.2.1, property 5, Ref. [13]) that the generalized inverse of V,

V^, is regularly varying with index p*=(l-2p)~^.

Furthermore, as g~V and is regularly varying we have

g^~V^. Then g admits an asymptotic inverse g^ which

is regularly varying with index p*=(l-2p)~^ where

0<p*<l. This means that g^(n/c^) ~ (l/c2)P g^(n) for aU oO, which implies that m ~ n/g^(n/c^) is equivalent to m ~ c2p*n/g^(n), 0<p*<l, oO.

Note that n/g^(n)eRVi_p*, 0<l-p*<l, that is, n/g^(n)6RV_2p/(i-2p> p<0.

Following Hall, Ref. [14], the next theorem shows that if p<0 something can be added to the results of theorem 4.2. THEOREM 4.3. Let FeDdiKGy) for some Ye E and assume that (4.3) holds with R(t)~tP, p<0. Then, for

intermediate sequences m=m(n) such that

(i) m = o(n-2p/a-2p)), n->oo, V m(Yn,e-Y) has

asymptotically a normal distribution with mean value zero and variance oHy,^)-

(ii) m c2^(^"2p)n-^P^(^-2p\ n-^oo, Vni(Yn,e-Y) has

asymptotically a normal distribution with mean value

bc(Y.9) and variance c\y,Q).

(iii) m/n-^P'(i-2p)_^^^ n-> oo, (n/m)''''(Yn,e-Y)

b(Y,e).

PROOF. Noting that R(t)-tP, for some p<0, implies

-2p/(l-2p)

t/g^(t)) ~ t , (i) and (ii) follow immediatly from

theorem 4.2. In what concerns (iii), from (4.4) and (4.5) we obtain the following representation

+ o((n/m)''R(n/m))

H(Y,e)

' 0-Y_l

since n~''/m^^^~''->0 and Tm^e ^ N(0,Va), n->oo for any

intermediate sequence m(n). Hence,

173
(^)-(V^,e-Y)=0

.D,-p(Am,9-e'^

e^HogG)

YH(y,9)

e-T(0-y_i)(_ioge)

•+o((n/m)-P(Am,0-e-^)

+ Op(l).

If the number of upper order statistics involved in the estimation of y is small the variance of the estimator is

large. But if we increase the number of upper order

statistics used in order to obtain a smaller variance, the estimator will have a bias different from zero. However in

this last situation the mean squared error of the estimator can be minimized if p<0 giving an optimal criterion to choose m. THEOREM 4.4. (Minimization of mean squared error) Let FeDdifCGy) for some ye E and suppose U has a second order behaviour with R(t)e RVp, p<0. Then, for sequences m=m(n)^«> and m ~ c^P*n/g^(n), with oO and

p*=(l-2p)~^, 0<p*<l, the mean squared error of the aP

estimator yn,e is asymptotically equal to

G^(y,e)/c^P* + c^(^-P*)b^(y,9)

n/g^(n)

If 0<p*<l the mean squared error is minimum for co=(pV(y,0)/(l-p*)b2(y,e))i/2 and if p*=l the mean

squared error is a decreasing function of c (the bias of the estimator remains constant but the variance decreases as m

increases).

PROOF. We have seen in theorem 4.2 and corollary 4.2 aP

that if m - c^P *n/g*"(n), Vm(yn,e-y) is asymptotically

normal with mean value bc(y,9) = cb(y,6) and variance

a^(y,0). Hence, the (asymptotic) mean squared error of

aP

Yn.e is

MSEoo(yn,8) = [oHy,e) + bj(ye)]/m

» [a2(y,e)/c2p *+c2(i-P*)b2(y,e)]/(n/g^(n)) = f(c)/(n/g^(n)).

* aP

If 0<p <1 the value of c that minimizes the MSEoo(yn,e)

is the zero of f(c) which is easily seen to be

co=[p*a2(y,e)/(l-p*)b2(Y,e)] If p* = l, we have

f(c)=a^(Y,6)/c^+b^(y,6) and hence the bias remains

constant and the variance decreases as c inaeases.

So if 0<p*<l we should consider the m largest

observations of the sample with m = [co^P*n/g'*~(n)] to

evaluate the estimator yn^e- The situation p =1 is more

complicated: it corresponds to slowly varying functions

R(t) and t/g^(t) and the above theorem does not give a

definite answer to the problem of choosing m.

5. Examples

In this section we illustrate the above results with the continuous distribution functions that are typically used in

statistical applications. In what concerns the differentiable domain of attraction of Gumbel distribution we will see

diat logistic distribution and Gumbel distribution itself have a second order regular variation behaviour while the Gamma(r), r^tl, and Normal distribution functions have a second order n-variation behaviour, as well as the

distribution function F(x) = l-exp(-x"), a>0, a^^l. For

the differentiable domain of attraction of GEV,y?iO, we

shall see that GEV(y?K)) and Cauchy distribution functions

have a second order regular variation behaviour.

Furthermore we consider the asymptotic normality of the estimator and determine the asymptotic bias for each one

of the continuous distributions considered as well as the

theoretical optimal value of m for the "polynomial rate"

distributions. EXAMPLE 1. For the logistic distribution we have

F(t) = (1+e-^)-^ te E, and U(t) = log(t-l). The function

tU'(t) = t/(t-l)€RVo is strictly decreasing with

lim tU'(t)=l and tU'(t)-le RV_i . We also have, for x>0,

^^^^^ = lOgX + (l-X-l-lOgX)t-l + 0(t-l ), t^oo.

Here ho(x) is a function of class (2) with ci=p=-l, C2=0

and R(t) = t"^ . Hence logistic distribution has a second

order regular variation behaviour. If m=o(n^'^) the

estimator has asymptotically a normal distribution with mean zero and variance a^(0,9)/m (theorem 4.2.(B)-(i)).

If m-dn^^^^ (i>0, Vrnyn,e has a bias equal to -d3/2(e-i)2/iog20 (theorem 4.2.(B)-(ii)) and the aP

asymptotic MSE of yn,e is minimum for m ~ don^^^

with do = [(l+e)/292(l-9)3]i^3 (theorem 4.4.).

174
EXAMPLE 2. For the Gumbel distribution we have

U(t) = -log(-log(l-t~^ )). The slowly varying function

tU'(t) = [(l-t)log(l-t"^ is strictly decreasing with

limtU(t)=l andtU'(t>-leRV_i. Also, for x>0,

t-x» ^^^^^ = lOgX + |(l-X-l-10gx)t-l + 0(t-l ). t-^oo.

It follows that ho(x) is a function of class (2) with

ci=-l/2, p=-l, C2=0 and R(t) = t"^ and hence Gumbel

distribution has a second order regular variation behaviour.

aP

If m=o(n2^^) the bias of Yn,e is zero (Theorem 4.2. (B)-

(i)) and if m ~ dn^^ , d>0, the bias is equal to -d3/2(e-l)2/(21og2eVm) (Theorem 4.2.(B)-(ii)). Moreover

aP

MSE (Yn.e) is minimum for m ~ don^^^ with

do = 2(1+9)'^ /[e2^3(i_e)] (nieorem4.4.).

EXAMPLE 3. For the distribution functions

F(x) = l-exp(-x'^), a>0, o.*\, we have U(t) = (logt)^'"

and tU'(t) = (l/a)(logt)^^"~^e RVq is a strictly increasing

function for a<l and a strictly decreasing function for a>l. One obtains, for x>0,

U(tx)-U(t) , l-g log^x., , I

. ^ tlJ'(t) " ^og^+~^ ^logt) ^+o((logt) t->oo.

Hence FeD(Go). Here ho(x) is a function of class (1) with

ci=(l-a)/a, C2=0 and R(t) = (logt)-i. It follows that

these distribution functions have a second order 11- variation behaviour. Note that ±tU'(t)e n(ai) with

ai(t) = (ll-al/a2)(logt)i/a-2. When m = o(log2n),

/—aP

^ D^Yn.e will be asymptotically normal with mean value

zero and variance a^(0,9); when m ~ dlog^n, d>0, the

distribution will have a bias equal to d^^(l/a-l) and the aP

MSE(Yn,e) is a decreasing function of d (the bias remains

constant and the variance decreases).

EXAMPLE 4. For the Cauchy distribution we have

F(t) = ^+^arctg(t), teE,

U(t) = tgf - 7) = ^ [1 - 1^ +

and tU'(t) = -{l+^l+^+o(t~^)]}, t^oo. One obtains,

for x>0,

= x-1 + f(2-x-i-x)r2 * 0(H),

and hi(x) is a function of class (4) with p=-2,

ci=-(2/3)Tc2, C2=0 and R(t) = f^. Moreover, U'(t) is a

strictly decreasing function with limU'(t) = tc^ and

t— U'(t)-7r^eRV_2. It follows that Cauchy distribution

function has a second order regular variation behaviour. For sequences m=o(n^^^ ) the asymptotic distribution of

/— aP , ^Di(Yn,e~l) is N(0,a''(l,9)) whereas for sequences m ~ dn^''^ the distibution will have a bias equal to d^^;c29(l-9^)/31og9. The mean squared error of

the estimator is minimum for m ~ don"^^^ where

do = [9(l+e3)/(47r494(l-e)3(l+9)2)]i/5_

EXAMPLE 5. For the Generalized Extreme Value

distribution, GY(x)=exp{-(l+7x)~^''^}, 1+yx>0, y^E,

we have U(t) = {[-log(l - (l/t))]-V- 1}/y, t>l, and

ti-ru'(t) = t-(V+i)(l - (l/t))[(-Iog(l - (l/t))]-(T^+i). For

Y<1, t^"UJ'(t) is a strictly decreasing function and for y>1

there exists to>l such that t^~1U'(t) is strictly increasing

for t>to. In both cases we have lim t'^iU'(t)=l. For x>0,

tU'(t) ~ Y 2\y-1 Y / ^

^'

t-^oo, Y'^l.

^^"^^^^ = x-1 + ^2-x-x-l )r2 + o(t-2 ), t^oo, ^1

.

Herehy(x) is a function of class (4) with ci=(y-1)/2,

C2=0, p=-l and R(t) = t"^ for y^I and with ci=l/12, C2=0,

p=-2 and R(t) = t"^ for y=1- Hence GEV distribution

function has a second order regular variation behaviour for any y^ E . If Y''! the conclusion (B)-(i) of Theorem 4.2. holds for sequences m = ©(n^^ ) whereas for sequences m ~ dn^^, d>0, holds conclusion (B)-(ii) of Theorem 4.2. with

bd(Y,e) = d3/2Y(e-i)(e-7+i-i)/[2(9-Y-l)log9]; the aP

minimum mean squared error of Yn,e is attained for m^don2^3 vvith do = [2(92Y+i+l)/(94(l-9)(l-9^i)2)]i/3. if Y=l and for sequences m=o(n'*^^) part (B)-(i) of theorem

4.2. holds whereas for sequences m ~ dn'*^^, d>0, part (B)-

175
(ii) holds with the bias bdd.G) = d5/2e(i_02)/(i2ioge).

Conclusion of Theorem 4.4. holds for sequences

m ~ don^'^ with do = [36(e3+l)/e4(l-e)3(9+l)2]i/5

EXAMPLE 6. For Gamma distribution,

+00

l-F(t) = z^, fs^-i e-^ds, r>0, r^l

,

r(r)J

and using the expansion

+00

js^-i e-^ds = e-V-i [l+(r-l +(r-l)(r-2)t-2+o(t-2 )],

t

t>0, one obtains, after some calculations,

r i-F(t+xfo(t)) ^x2 „ l-F(t)

-e ^J/P(t) = ±^^

with the plus sign for r<l and the minus sign for r>l,

fo(t) = (1-F(t))/F'(t) and p(t) = +(r-l)t-2. By theorem

A. 10 in Ref. [7] this is equivalent to ±tU'(t)Gn(a)

with a(t) = fo(U(t)).p(U(t)) = tU'(t).p(U(t)) which is

equivalent (theorem A.5, Ref. [7]) to ^M=mi ^ iogx-(r-l)^(t))-2 + o((U(t))-2), t-.c«,

for x>0, and we have to fmd an asymptotic expression for U(t). As l-F(t) ~ F(t) one obtains U(t) - (r(r)ti-ret)^ ~ ~ log(t/r(r)). Hence, for x>0,

logx - (r-1) !^(logT^ + o((logt)-2).

with ho(x) a function of class (1) with Ci=r-1, C2=0 and

R(t) = (logt)~^ and gamma distribution has a second order

n-variation behaviour. For sequences m=o(log'*n),

j aP

^niYn,9 has asymptotically a normal distribution with mean value zero and variance a^(0,9), and for sequences

m - dlog'*n, d>0, the asymptotic distribution has a bias equal to d'^^Cl-r); as d increases the bias of the estimator

remains constant and the variance decreases. EXAMPLE 7. For normal distribution,

+00

l-F(t) = je-s^^ds

V 2tc t and using the expansion

400

Je-s^^ds = e-^^^rH\-r^+o{r^)}, t-^00,

t one obtains

with fo(t) = t-i {l-t-2+o(t-2)} and P(t) = t'^. This is equivalent (theorem A.IO, Ref. [7]) to -tU'(t)en(a) with

a(t) = tU'(t).p(U(t)) and this statement is still equivalent

(theorem A.5, Ref. [7]) to ^M^^ = logx - ^

(U(t))-2 -H o((U(t))-2), t->oo,

for x>0. As l-F(t) -F(t)/t we have U(t) ~ C^nte^^'^)^ ~

~ V2(logt)i^2 Hence for x>0,

5Mt) = log, _ loS^

(logt)-. . „((,og,)-. ), t^,

with ho(x) a function of class (1) with ci=l/2, C2=0 and

R(t) = (logt)~^ and normal distribution has a second order

ri-variation behaviour. For sequences m=o(log^n),

/—aP \myn,e has asymptotically a normal distribution with mean zero and variance a^(0,G). For sequences m ~ dlog^,

d>0, the distribution has a bias equal to -d^'2/2 and as d

increases the bias of the estimator remains constant whereas the variance decreases.

6. On choosing the parameters

The variance of the generalized Pickands' estimator,

oHy,Q), does not have the same behaviour, as a function

of 9, for all real y. Thus it is not possible to choose a value of 9 in ]0,1[ that minimizes the variance for any real

y. Anyway a value of 9 can be choosen in order to improve Pickands' estimator. We can look for the value of 9 that minimizes the variance of the estimator when y=0

because of the central role the Gumbel distribution plays

in extreme value theory. It is easily shown that when F is aP

in Ddif(Go), varCVn.e) has a minimiun for 9o the unique

zero of the function s(9) = (2+log9)/92-2 in ]0,1[, i.e., aP

9o«0.14. The estimator yn,o.i4 is asymptotically more

efficient than Pickands' estimator for (approximately)

176
Y<1.3. In what concerns the fraction of the sample to be

used in the definition of the estimator, it does not seem

possible to find an optimal criterion for choosing the value of m independently of the underlying distribution. However the estimator is expected to present always the same kind of behaviour: a great variability for small

values of m, a more or less constant value for moderate

values of m and, after, a significant increase of the bias for great values of m. In practical applications, estimates of y

for the different values of m should be calculated and then be considered for y the value more or less constant corresponding to the relative stability phase of the

estimates.

Acknowledgements. I am indebted to Professor J. Tiago de Oliveira for suggesting me the study of

generalized Pickands' estimator and for many interesting

and valuable discussions on extreme value theory. I would

also like to thank Professor Ivette Gomes for her

important comments and encouragement.

Ph.D. thesis, Erasmus University, Rotterdam, 1991.

[9] Pickands ni, J., Statistical inference using extreme

order statistics, Ann. Statist., 3 (1975), 119-131.

[10] Cooil, B., Limiting multivariate distributions of

intermediate order statistics, Ann. Probab., 13 (1985),

469-477.

[11] Cooil, B., When are intermediate processes of the same stochastic order?, Statist.&Probab. Letters, 6 (1988),

159-162.

[12] Tiago de Oliveira, J., Extreme values and tail estimation. In: Order Statistics and Nonparametrics;

Theory and Applications (ed.: I. A. Salama), Alexandria.,

1991, pp 1-16.

[13] De Haan, L., On Regular Variation and its Application to the Weak Convergence of Sample

Extremes, Math. Centre Tracts 32, Mathematical Centre,

Amsterdam, 1970.

[14] Hall, P., On some simple estimates of an exponent

of regular variation, J. Roy. Statist. Soc. Ser. B, 44

(1982), 37-42.

REFERENCES

[1] Gnedenko, B.V., Sur la distribution limite du terme maximum d'une s6ries aldatoire, Ann. of Math., 44

(1943), 423^53.

[2] Von Mises, R., La distribution de la plus grande de n

valeurs, (1936). Reprinted in Selected Papers II. Amer.

Math. Soc, Providence R.I. (1954) 271-294.

[3] De Haan, L., Slow variation and characterization of

domains of attraction. In: Statistical Extremes and

Applications (ed.: J. Tiago de Oliveira), Reidel, Dordrecht,

1984, pp 31-38.

[4] Pickands IH, J., The continuous and differentiable domains of attraction of the extreme value distributions, Ann. Probab., 14 (1986), 996-1004.

[5] Geluk, J. L. and de Haan, L., Regular Variation,

Extensions and Tauberian Theorems, Math. Centre Tracts

40. Centre for Mathematics and Computer Science,

Amsterdam, 1987.

[6] Omey, E. and Willekens, E., n -variation with

remainder, J. London Math. Soc, 37(2) (1987), 105-118.

[7] Dekkers, ALM. and de Haan, L., On the estimation

of the extreme-value index and large quantile estimation, Ann. Statist., 17 (1989), 1795-1832.

[8] Dekkers, A. L. M., On Extreme-Value Estimation,

177
I
Normal Sample Range: Asymptotic Distribution,

Approximations And Power Comparisons

Rukhin, A.L.

UMBO, Baltimore, MD

Abstract

The largest interpoint Euclidean distance gives a quick estimator of the

unknown variance of a two-dimensional normal sample and also provides a

short-cut test statistic for this parameter. Its asymptotic distribution and

bounds for moderate sample sizes are discussed. In particular the power

comparison of the optimal test and the test based on the sample range is reported.

1 Asymptotic Distribution of the Sample Range

This work was inspired by a quality

control problem which arises in the

manufacturer's testing of handguns. A

handgun is placed in a vice and fired ten

times at a target with a grid on it after which the largest interprojectile distance is determined. If the distance ex- ceeds 4 inches the gun is rejected. The

advantage of this method is that this

determination is an easy task whereas

the calculation of the sum of squares

needed for the optimal test about the

dispersion of projectiles is time consuming and not always feasible.

Thus the bivariate sample range is an

179
important statistic in gun quality control and it also appears rather naturally

in other accuracy related problems of

vector observations. Wilks obtained by

the Monte Carlo method the first four

moments of the bivariate normal sample range for some values of the sample size n, which are reproduced in [1] where also a chi-approximation is suggested.

In general the sample range could

be used for detecting outliers (cf. Ch.

9.3 of [2]) or to provide a quick estimate of the dispersion. The monograph

of Grubbs [3] discusses further statistical measures of precision which are

also summarized in Section 7.5 of ref

[4]. where this statistic is described

as "intriguing". Despite obvious interest in the distribution of the bivariate normal sample range its asymptotic form has not been determined until recently by Matthews and Rukhin

[5]. Let A'i,X2,... be independent

A;-dimensional normal random vectors

with zero mean vector and the identity

covariance matrix. The sample range R

is defined as the largest interpoint distance between the first n observations

R = Rn = max \Xi — Xj\.

l<t<7<n

The exact and asymptotic distribution of R is well known in the special case k = I. Namely for any positive r with = log log n

Pr(y2 log n[R - 2^2 log

n

+ log 47r

, < r] V2l^ J - ; /oo

exp{-e'-' -e-')dt, (1)

-oo

i.e. the asymptotic distribution of the

range is the convolution of the limiting distributions for the extreme order

statistics.

A related result on the asymptotic

behavior of random points with specified nearest neighbour relations was obtained in [6]. Theorem 1 For k > 2 and positive r

lim Pr{R<2 n—t-oo 21og n ^{k - 3)/2n

7 +1371 + a + r

j

lim Pr(J2logn\R - 272 log

n

n—i-oo \ ' L '

0.5(A; - 3)/2n + /gn -f a V21ogn = exp(— e"'").

< r

Here

and

a = a(k) -- log

Isn = log

{k-l)2^

r(A;/2)7ri/2

Let C-n denote the cardinality of the set

{(z, j) : 1 < 2 < i < n, \X,-X,\ > 2ri},

Pr i^R < 2[21og n-hn- logAir + r] ' j

i.e. the number of exceedances by the

180
interpoint distances of the given level

2ri where

ri = [21ogn+i(A;-3)/2n+/3n+a+r]^/l

(2)

The equivalence of the events C„ > 0

and R > 2ri, and the following more

general result imph' Theorem 1. Theorem 2 As n tends to infinity, Cn

converges in distribution to a Poisson

random variable with parameter e"'".

The proof of Theorem 2 is based on

the relationship of exceedance count Cn

to the Poisson clumping heuristic ar- gument. It turns out that the sample

size n can be replaced by a Poisson

number of points with mean n without affecting the asymptotic distribution. This fact allows to transform

the problem into the one about a Pois- son process on the A;- dimensional Euclidean space possessing intensity function n{27:)-^l'^exp{-\x\^ 12) with independent number of points in disjoint re- gions of space. A coupling argument

establishes the asymptotic equivalence.

If ri is given by (2),

r2

= [21ogn+-(A;-3)/2n+-|—|-2(a-hr)]^/2

rs = [2 log n + {k- 2)/2n + 2/3n]^/^

and

ro - 2ri - r2

then one can show that radii ri , r2, rs, r4

and the points leading to exceedances

are in a narrow annulus at (2 log ny^'^ +

0(/2n(log n)"-^/^). (These values are

slightly different from the ones given in

[6] where a superfluous factor 2 appears

in the denominator on the third line on p. 456. The author is grateful to

H. Henze who noticed that the original

radii were incorrectly specified.) There

are no points with \Xi\ > rs with probability tending to one and that there are no pairs of points, whose lengths are between r2 and rs, with distance between

them exceeding 2ri. Intuitively, points

in this range are sparse enough so that

their angular separations are not likely

to be close enough to tt (which would

lead to a sufficiently large interpoint

distance). It can be proven that large

interpoint distances with both points

within the radius r2 and between tq and

r2 are also nonexistent with probability

tending to one. Finally a coupling ar- gument shows that the number of interpoint distances exceeding 2ri, asymptotically has a Poisson distribution.

Intuitively, a vector Xi with an ex- ceptionally large norm could lead to a clump of exceedances of 2ri when com-

'bined with vectors of large length and

nearly opposite direction. Indeed it looks that this possibility prevents the

moments of Cn from converging to the

moments of the limiting Poisson distri181
bution. For example when k = 2

lim ECn = oo.

2 Lower and Upper

Bounds

In this section we discuss some approximations to the distribution function Fn{r) = P{Rn < r) of the

bivariate normal sample range i?„

based on the random sample Xi =

{Zi, I'l), . . .

, Xn = (Zn, In)-

Notice first of all that

n

Rn = max max

0<e<2x l<i,j<n ( — ) cos B

+(y;- - y})sin^ = max max

0<e<27r l<j,j<n

Zj cos d -\- Yi sin Q)

{Zj cos^ + Yj smO) (3)

Therefore

Rn > max"^ max Ut — min Ui,

max Vi — min ¥{ \ l<i<n l<i<n J where t/, , , = 1 , . • .

, n are independent standard normal random variables.

As was mentioned in Section 1 the

distribution of the scalar normal sample

range is well known. Its distribution

function Gn has the form

Gniu)) = P ( max Ui — min Ui < w

\l<i<n l<i<n

/CO ,

-oo (4)

with the limiting distribution specified

in(l).

Thus

Fn{r) < Gn{rf

and better upper estimates can be derived from (3) by considering a larger

nuber of angle 9 values.

Let pn denote the radius of the

smallest circle containing the twodimensional normal sample. Then

Rn < 2pn. (5)

The distribuion of /?„ is known (see refs

[8,9]). Namely

Hn{z) = P{2pn < Z)

= n (1 — e -72/8 n-l

(n-1) 1- _,2

(6)

In other terms p^ has the distribution of

the (n — l)-th order statistic in a sample of n independent distributed

random variables. Thus (5) provides a sharper bound than the maximum of

such a sample which corresponds to inequality Rn < 2max{

By combining the inequalities above

one can formulate the folllowing result.

182
Theorem 3 With Hn and Gn defined

by (6) and (4) one has for all positive r

Hn{r) < Fr^ir) < Gl{r).

Bounds of Theorem 3 also lead to

bounds on the moments of Rn- For instance

ERl<iEpl = 8[^{n-\-l)-^{2)]

1 <8 log(n + l)-l + C2(n + l)_

and for sufficiently large n

^ V2 log n

Here ^ denotes the log-derivative of

gamma-function so that ^(2) = 1 — C = I — 0.5772... These bounds turn

out to be reasonably tight for moderate values of n. However the question about the con- vergence of the moments of Rn to these

of the extreme value distribution re- main open. For instance it is not known

l2n

The approximations of Theorem 3 show

that this limit is between —4 and 0.

Figure 1. The distribution

functions of F„ (solid line), Hn

(dashdotted line) and (dotted

line) for n = 10.

Figure 2. The distribution

functions Fn (dotted line) and En

(solid line) for n = 10.

Figure 3. The distribution

function of (dotted line) and

En (solid line) for n = 10.

183
10 15 20 25

Figure 1 show for n = 10 the distribution functions of the lower and upper bounds Hn and along with the

Monte-Carlo simulated empirical distribution function Fn for 10000 repetitions. Figure 2 shows Fn and the distribution function En of the best fitted

extreme value distribution,i.e.

3 Power comparisons

Let X, = iZuY,),...,Xn = {Zn,Yn)

be a random sample of two-dimensional

random normal vectors with independent coordinates (Zi,y^) with zero means and the same unknown variance

a^. As indicated in Section 1 the quality control problem for handguns leads

to the hypothesis testing Hq : a < ao

30versus the alternative Hi : a > gq. The optimal (uniformly most powerful)

test has the critical region of the form

{52 > xj(2n - 2)} where

En{r) = exp{-e(/?-r)/afor a and ^ chosen so as to match the

first two moments. Figure 3 provides

the same graphs for the distribution of

i?^. Our simulations suggest that the

distribution of the squared range al- lows a better approximaton by an ex- treme value distribution.

The fact that extreme value distribution approximations are often more

accurate for squares of extreme order

statistics is actually well known in the

classical asymptotic theory of normal

order statistics (see [10,11]).

Z,-Z} +iY,-Yy

(2n - 2)al

Here Z, Y are the coordinatewise sample means and Xa("^) denotes the critical point of distribution with m degrees of freedom.

In the situation mentioned in Section

1 a handgun is rejected if in consequtive

10 shots the largest inertprojectile distance exceeds 4 inches. This procedure

corresponds to a test of level 0.05 for

cfq — .79. Indeed the Monte- Carlo simulation for n = 10 gives 95-th percentile

of the distribution of thwe sample range

to be about 5.14.

The results of numerical comparison

of powers of tests based on Sn and Rn

are given in Figures 4-6 for n = 5, 10

and 15. The power function of the test based on the sample range never falls below 65% of the power of the optimal

184
test. Figure 4. Power function of the

optimal test (dashed line) and

the test based on the sample

range (dotted line) for n = 5.

Figure 6. Power function of the

optimal test (dashed line) and

the test based on the sample

range (dotted line) for n = 15.

Figure 5. Power function of the References

optimal test (dashed line) and

the test based on the sample

range (dotted line) for n = 10.

[1] T. Cacoullos and H. DeCiccio. On

the distribution of the bivariate

range. Technometrics, 9:476-480,

1967.

[2] V. Barnett and T. Lewis. Outliers

in Statistical Data. Wiley, New

York, 1984, 2nd ed.

[3] F. E. Grubbs. Statistical Measures

of Accuracy for Riflemen and Missile Engineers. Edwards Broths,

Ann Arbor, 1964.

[4] H. A. David. Order Statistics. Wiley, New York, 1981, 2nd ed.

185
[5] P. Matthews and A. L. Rukhin.

Asymptotic distribution of the

normal sample range. Ann. Appl.

Probab, 3:454-466, 1993.

[6] N. Henze. On the fraction of ran- dom points with specified nearestneighbour interrelations and degree of attraction. Adv. Appl.

Prob., 19:873-895, 1987.

[7] H. Daniels. The covering circle of a sample from a circlar normal distribution. Biometrika, 39:137-143,

1952.

[8] E. J. Halteman. The Chebyshev

center: a multidimensional estimate of location. Journ. Statist.

Planning Inf., 13:389-394, 1986.

[9] J. B. Haldane and S. D. Jayakar.

The distribution of extremal and

nearly extremal values in samples from a normal distribution.

Biometrika, 50:89-94, 1963.

[10] S.B. Weinstein. Theory and applications of some classical and generalized asymptotic distributions of

extreme values. IEEE Trans. Inf. Theory, 19:148-154, 1973.

186
Estimation Of Extreme Sea Levels At Major

Ports In Korea

Shim, J.S., Oh, B.C. and Jun, K.C.,

Korea Ocean Research & Development Institute, Seoul, Korea

The design of coastal structures requires knowledge of the probability of extreme sea

levels, as well as of extreme wave heights for safety. Two methods for computing

extreme sea levels, the annual maxima method and the joint probability method, are examined for major ports (Incheon, Cheju, Yeosu, Pusan, Mukho) in Korea. The annual

maxima method estimates the extreme sea levels from three different probability

distributions of Gumbel, Weibull and generalized extreme value(GEV) using the least square method(LSM), the conventional moment method(CMM) and the probability

weighted moment(PWM) method, respectively.

The results show that the extreme sea levels estimated by the Gumbel distribution or

the least square method appear, in general, higher than those calculated by other

distributions or methods. The extreme values estimated by the extreme probability

method are approximately 5-lOcm lower than the values estimated by the joint probability

method.

1. Introduction

The rise and fall of sea level is caused by

the repetitive combination of astronomical tide

and storm surge. The Office of Hydrographic

Affairs in Korea has observations of the sea

level since 1960 at the major ports of the

country.

The extreme sea levels obtained from

relative long-term tidal data play a very

important role not only in planning the overall

layout of coastal structures, but also for fixing

the positions of the intake pipes of nuclear

power plants. The information about the

extreme sea levels is especially important in Korea where there are large tidal ranges in the

southern and western coasts of the country.

The best way to design a structure must be

based upon the proper analysis of field data

obtained at the possible construction site, together with the appropriate consideration of the

functional and financial constraints and the life

time of the structure. In general, we often

perform the preliminary design of the structure through the extreme statistical analysis of

hindcast data since we may not frequently

accumulate enough observated data.

There are two kinds of estimation methods

of the extreme sea level, that is, the annual

maxima method and the joint probability method.

The first method makes use of the distribution

function of maximum sea level.^"' Therefore for a place of interest, the annual

maximum for each year is extracted from hourly

observed sea level and is used to estimate the

parameters of the probability distributions. The

latter method calculates the extreme sea levels

by convoluting probability density functions of

the tide and surge components on the

assumption that the two components are independent of each other.^^^' In this study, the extreme sea levels are computed by the two methods at major ports

(Incheon, Cheju, Yeosu, Pusan, Mukho) of the

187
that the number of hours in a year is large

enough for the asymptotic approximation to hold.

Table 1 gives the expressions for Fix)

defining the different distributions considered

here, and also includes expressions for their means and variances. We estimated the

parameters of the distributions by means of

LSM, CMM and PWM.

2.1. Least Square Method (LSM)

This method is to provide a straight line fit to the data when it is plotted on a pertinent

probability paper. This gives a slope(a) and an

intercepted) of the best-fit line y = ax + b in terms of the coordinates(x,, y,) of all data points.

The corresponding estimated values of the

distribution parameters, if required, may then be

obtained from the slope and the intercept by the

expressions given in Table 2. In estimating the parameters by LSM,

plotting position is necessary. In this paper, the

Gumbel plotting position is used as follow^; F(x,)=^ (1)

where / denotes the rank of data, with i = 1 for

the smallest value and n for the largest value,

and n is the number of data.

2.2. Conventional Moment Method (CMM)

In this method, parameters are estimated

Table 1. Asymtotic probability distributions function

Distribution Range Cumulative probability Mean Variance

Gumbel

- CD<x< 00 - 00 < £ < CD o<e< 00 exp[-exp{-(

g ))]

£ + T*9

(==£ + 0.589)

6

^

(=*i.64e2)

Weibull

E<X< 00 o<e< 00 0<a< CO l-exp{-(

Q ) } £+er(i.-^)

9^{r(i.^)

GEV"

£<X< 00 0<8< 00 - 00 < £ < 00

j_

exp[-{l--|-(x-£)}

]

£ + e{l-r(l+a)}/a 82{r(l+2a)

-YHl-a)}/a^

* : T is Euler's constant equal to 0.5772 ** : if a=0, GEV equals Gumbel

188

country with relatively long-term observation

data. The annual maxima method estimates

extreme sea levels from the three different

probability papers of Gumbel^"\ Weibull^^^^ and

generalized extreme value(GEV)^^^', each of

which is prepared by applying three different

methods for estimating parameters; tlie least

square method (LSM), the conventional moment

method(CMM) and the probability weighted

moment(PWM) method. The joint probability

method, compared with the annual maxima

method, is more useful when only few years

observations of sea level are available. In the

annual maxima method, however, long-term data

is needed. In this study, the long-term data is used in both the joint probability method and the

annual maxima method in order to compare their

results under the same condition. Considering

the data requirements of each method likewise,

we analyze and compare all the results estimated

by the above methods.

2. Annual Maxima Method

The annual maxima method has been a

classical method for analyzing extreme values,

applied to sea level estimation since Ref. [1], [2]. In particular, this was the method used in the

comprehensive study by Ref. [5]. The assumptions made in using this method

are namely that hourly sea level heights are (1)

independent, (2) identically distributed and (3)
Table 2. Scale relationships for probability distributions

Distribution Abscissa scale(x) Ordinate scale(y) Slope(a) Intercepted)

Gumbel

Weibull

GEV

X

ln(x-£)

X

X

-ln[-ln{F(x)}]

ln[-ln{l-F(x)}]

1 [-ln{l-F(x)}] °

[l-{-lnF(x)}"]/a

1/9

a

1/8

1/8

-£/8

-a In 9 -£/9

-£/9

from the nioment of the probability density

function for the distribution. The moments

which are the first, second or third ones, are estimated from the sample. This method often

leads to an acceptable model, since the lower

moments have the stronger influence on the

shape of the distribution. ^^"^^ The estimated

values of the parameters are expressed in terms

of X, ~)? and "x^ as indicated in Table 3. Here

X, ~?

and 3x" are obtained directly from the

data and are defined as follows; - \ ^ 1 2 -3 1 V 3 x= — 2-,Xi, X = — 2->Xi , X = — ZuXi (2)

Because the Gumbel distribution has two

parameters, these can be easily obtained as given in Table 3. The Weibull and the GEV

distributions involve three parameters, and of

these, a is first estimated by equating the

skewness( V3) of the sample to those of the

model. The remaining parameters can then be

estimated from the first and second moments.

can be expressed as follows;

Ja M /,. ^ P-3xI^+2(;c)^

V3 = U3/U2 = 3 (3)

in which U2 and Us are the second and third

central moments of the distribution. Eq. (3) is replaced by the function of a;

Vp = ( r( 1 + -S- ) -3r( 1 + 4- ) r( 1 + 4- ) +

(4)

2r2(i + 4-)|/{r(i + -|-)-r'(i + 4-)|

2.3. Probability Weighted Moment (PWM) Method

A new class of moment, called probability

weighted moment, was introduced by Ref [15].

It was indicated to be of potential interest for

distributions that may be written in inverse

form, that is, if X is a random variable and F is the cumulative distribution function for X, the

value of X may be written as a function of F ' X = xiF). Reference [15] defined a probability

weighted moment as;

Table 3. Parameters of distributions as estimated by conventional moment method

Distribution

Estimated parameters

a £

Gumbel ^ [x -{x) ) X- y8

Weibull Vr= /(a) i ~?-Cxf 1

12 x-Br(i + i/J)

i r(l + 2/a)-r2(l + l/a) J

GEV V3"= /(a)

1 f X -{x)

12 3c-aB{l-r(l + l/a)} a \ r(l+2/a)-r'(l + l/a) j

189
^xmVFKl-F)" dF

(5)

where /, j, k are real numbers.

If an interesting probability distribution

function is substituted for F, it is possible to make an integral of Eq. (5) about the fixed set

(/, 1 k), and the result of the integral is expressed as the function of F's parameters.

Therefore as we calculate PWM about as many

sets (/, j, k) as the unknown parameters, we can

get simultaneous equations for these unknown

parameters.

To practice the above method, we have to decide the set (/, j, k) first of all. In the case

of PWM method, since either /=!, y= 0 or /=!,

/c = 0 uses, X certainly has the first power and

either F or \ - F \s excluded. That is, the

following conventional equations are adopted;

(fc) = Mi,o./c= Cx{l-F)''dF

Jo

j = Muo = j^xF'dF

(6)

(7)

There are a lot of differences between the

above equations and the conventional moment

below.

r = X"" jix)

J- oo dx (8)

One of the differences is that the Eq. (8) has

the operation of the rth power about x so that

the observation error or abnormal values are

amplified as the power gets higher, but since

PWM of Eq. (6) and (7) places its operation in cumulative distribution function F, the sampling

errors or abnormal values become smaller. This

effect is, however, estimated from the definition

of the moment merely, and there has been no

flfil definitive evaluation of this approach.

If j and k are nonnegative integers, then

Ik^ MiM = g ( j) {-ly MuQ

(9)

In the special case where /, j and k are nonnegative integers, Mi,j,k is proportional to Fix 'j*i,k*ri), the Ith moment about the origin

of the ij+ l)th order statistic for a sample of

size k + j+ 1. More specifically,

Mij,k =B(;+l,fc+l) E [X'rik*j.i] (10)

where 5( •

, • ) denotes the beta function. For

j = 0 and / = 1 the convention

M(k) = Mi.o.k = B{lMl)F[XiMi] (11)

is adopted. M{k), unbiased estimate of M{k)

from a sample size of n and where k is a nonnegative integer, is obtained as follows;

Y n-k n f^i

(12)

n-i

k

in-l

And also, for k = 0 and / = 1, the Eq. (10)

becomes

Mj = Muo = EiXj.i.j.i] (13)

To get the unbiased estimate Mj of M>, we

need the EiXru^i), the first moment about the

origin of the (y+ l)th order statistic for a sample

of size In drawing randomly j+lin> j^l)

of the sample xi, xz, —, Xn, the probability of

which the maximum value is Xi is Therefore EiXj*i,j*i) is

' ^[^'•''•1= »ffi,l;^'('';-V(y"i) '1^' Mj can be estimated from the Eq. (13) and

(14) as follows;

J^Y.- {i-i)U-2)-(i-i) /•>-,^ n fH (n-l)in-2y-(n-j) ^J-^'

Mj= (15) n , = 1 (y=0)

When the sample of size n is arranged from

xi to Xn in ascending order, either an estimate

M (k) from the Eq. (12) or Mj from the Eq.

(15) can be made. And then the solutions of

PWM are obtained by substituting them into the

simultaneous equations of the parameters.

The probability weighted moments, M/,oa or

Mi,j,o of three distributions are given in Table 4, and the parameters of each distribution are shown as Mj and M(fe) in Table 5. From Table

3 and Table 5, Gumbel's parameters are defined

explicitly as the functions of both conventional

moment and PWM. On the other hand, those of

Weibull and GEV can be explicitly expressed as

190
the function of PWM only, not as conventional

moment

Table 4 Expressions of probability weighted moment

Distribution

Gumbel

Weibull

DEV

PWM Mi,i.k (real i/c >0)

Mifi,k =Mik)= +

(l + /c) ^^^^

Mi,j,o = Mj^ e^e{l-(/Vl)'T(l^a)}/a

Table 5. Parameters expressions of probability

weighted moment

Distribution Parameter PMWM(k),Mj

Gumbel

£ Mo-tS

(2Mi-Mo)/ln(2)

£ =0 0

M(0)/r[ln{M(0)/M(i)}/ln(2)]

a ln(2)/ln{M(o)/2iW(i)}

£7^0

4(M(3)M(o)-Ma))

Weibull 4M (3)+M(0)-4M (1) % M(o)-£

a

, f M(o)-2M(i) 1

a ln{ (Mo-2Mi)/2(Mi-2M3)}

In 2

DEV B a(2Mi-Mo)/{r{l+a)(l-2'"))

£ Mo-e{l-r(l+a)}/a

3. Joint Probability Method

In this method, the hourly observed data is separated into mean sea level, tide and surge

components. Then the probability density

functions of the tide and surge components are convoluted to obtain the probability of a

particular sea level, incorporating return period

levels. This method has advantages over the

annual maxima method in that the method can

estimate the extreme sea levels even with

short-term observed sea level data and can get

the low extreme sea level as well. In addition,

this method can evade difficulties for

establishing a suitable distribution function and

for estimating unbiased parameters in the annual

maxima method.

Any instantaneous value of sea level ^it)

measuured from a defined datum may be

considered to be a sum of three independent

components; mean sea level Zoit), tidal level Xit) and residual or surge level Yit).

C(f) = Zoit) + Xit) + Yit) (16)

Annual mean sea level for a particular year can

be determined from hourly observed data, and is removed from the data. However, annual mean

sea levels are not constant. The tidal component of the sea level data is directly or indirectly affected by astronomical

forcing. It is also removed from the observed

sea level data expressed as the finite sum of

harmonic constants which have the following

form;

n Xit)= T.fnHnCOs[ant+ iVn+Vin)+ gn] (17)

n= 1 where Hn is the amplitude of the nth

constituent. On is its angular speed defined

astronomically, Vn is its equilibrium tidal phase

at t = 0, gn is the phase lag of the constituent on

the equilibrium tide, and fn and Un are the nodal

corrections. Removal of the tidal component as above does not require an excessive length of

record as a satisfactory tidal analysis can be

obtained from even one year observation.

Once both the tide and the mean sea level are removed from the observed data, only the

surge or non-tidal component remains. Over a

sufficiently long period the surge is a random

variable. Obviously over a short period like a month, however, very few surges are likely to occur and produce random phases.

If the tide and surge components are then

independent each other at any time t, the sea

level relative to the mean sc^ti level, i.e. w = ^it) - Zoit), may be regarded as the sum

of two independent components x = Xit) and

y = Yit). Thus if the probability density

functions of the tidal and surge components are

ftix) and fsiy) respectively, then the probability

191
density function fiw) of w is

fiw) = f ftix)fsiy)dy

J- CO = r Mw-y)fsiy) dy

J- 00

(18)

The onussion of dependence on time, t, when

replacing Xit) by x etc. implies an assumption

of stationaiity for the series Xit) involved. The

hourly predicted tide series is generally

considered to be stationary, but the hourly

residual series is, to some degree, nonstationary

since seasonal and meteorological effects like storm surge will give rise to series of residual

not randomly distributed in time.^^°^ The probability of exceedance of a particular sea level t) may be evaluated from the

corresponding cumulative distribution function

Fc(Tl) = Prob(C ^Ti) defined as follows;

dw

(19)

F[(.w-y) fsiy) dydw

n J-oo

The alternative form;

l-F;(Ti) = l- f Fciy\-y)fsiy)dy (20)

J- oo where Fti • ) is the cumulative distribution

function of tide. Reference [8] suggested that the return period in year of a particular sea level t\ is expressed as;

i?p = l/[{ 1 - F?(T1)} X ] (21)

where X is the number of data used per unit

time, and has 1.0(annual maximum value in a

year), 8766(average number of hourly values in a year) for the annual maxima method and the

joint probability method, respectively.

4. Evaluation of Extreme Sea Levels

The Korean Peninsula is enclosed by the

Yellow(West) Sea, the South Sea and the Sea of

Japan(East Sea). In the Yellow Sea, the bottom

topography is very flat and the average water

depth is about 40m. Its tidal range is very

large and increases to the north, reaching about

8.0m at Incheon. In the South Sea, the average

water depth is about 100m and the bottom is fairly flat The tidal range increases to the

west recording about 3.0m at Yeosu. The Sea

of Japan whose average depth is about 15(X)m

has monotonous shorelines and very steep

shelves. The tidal range is about 0.3m only and

increases to the south, reaching about 1.2m at Pusan.

The lengths of records are 29 years at Incheon, 27 years at Cheju, 25 years at Yeosu,

30 years at Pusan and 22 years at Mukho,

respectively. Their locations are shown in Fig.

1. Editing tidal data is necessary to eliminate

erroneously recorded data prior to evaluating the

extreme sea levels. The editing method chosen

to examine the sea level records fundamentally

consist of plotting the observed sea level and

the surge as a function of time and of

examining the plotted values by eye for

detecting errors marked as irregiilarities and

spikes. The errors are then corrected by

referring to the original tide gauge chart.

CHEJ

122° 124° 126 128^ 130^ 132^

Fig. 1 Location map

The extreme sea levels are evaluated by the

above-mentioned various methods, and are expressed in Table 6. The result of the annual

maxima method at Incheon is shown in Fig. 2. In this figure, the lines representing CMM and

PWM are plotted in the probability paper after estimating unbiased parameters and have no

connection with plotting position.

As given in Table 6, the extreme sea levels

calculated by the joint probability method are

192
Table 6. Extreme sea levels with return periods estimated with various methods

(a)Incheon, (b)Cheju, (c)Yeosu, (d)Pusan, (e)Mukho

(a) Incheon (Sea level relative to defined datum, Unit : cm)

Return

period

(yr)

Annual maxima method

Joint pre met

)bability

lod

Remark

Gumbel Weibull GEV High

extreme

sea level

Low

extreme

sea level

Max.(min.)

LSM CMM PWM LSM CMM PWM LSM CMM PWM observed

29 986.0 982.1 984.0 980.3 978.9 978.9 980.9 978.3 979.6 982.6 -128.5

984.0

(-102.0)

50 991.8 987.2 989.5 983.3 981.9 981.7 984.0 982.3 982.6 987.6 -135.3

100 999.2 993.8 996.6 986.8 985.3 985.0 987.3 985.7 985.9 993.9 -143.6

200 1006.2 1000.3 1003.5 989.9 988.4 987.9 990.2 988.6 988.7 1000.2 -151.4

300 1010.9 1004.1 1007.6 991.6 990.1 989.5 991.7 990.1 990.1 1003.8 -155.8

(b) Cheju

Return

period

(yr)

Annual maxima method

Joint FYobability

method

Remark

Gumbel Weibull GEV High

extreme

sea level

Low

extreme

sea level

Max.(min.)

observed LSM CMM PWM LSM CMM PWM LSM CMM PWM

27 320.3 318.2 318.6 319.9 317.9 317.4 319.9 317.9 318.3 330.5 -52.4

324.0

(-48.0)

50 323.8 321.3 321.7 322.8 320.3 319.5 323.1 320.6 321.1 334.3 -55.7

100 327.6 324.6 325.2 325.8 322.8 321.6 326.6 323.4 324.2 338.7 -59.6

200 331.5 328.0 328.7 328.8 325.1 323.6 330.0 327.6 328.9 342.9 -63.5

300 333.7 330.0 330.7 330.4 326.4 324.7 332.0 328.6 330.1 345.3 -65.7

(c) Yeosu

Return

period

(yr)

Annual maxima method

Joint probability

method

Remark

Gumbel Weibull GEV High

extreme

sea level

Low

extreme

sea level

Max.(min.)

observed LSM CMM PWM LSM CMM PWM LSM CMM PWM

25 419.3 416.5 417.6 418.5 415.5 417.6 419.2 415.8 417.5 423.5 -54.9

416.0

(-57.0)

50 424.5 421.0 422.4 422.7 418.6 421.7 424.3 419.0 422.3 428.5 -58.4

100 429.6 425.4 427.2 426.6 421.4 425.6 429.4 422.0 427.0 434.2 -61.9

200 434.7 429.9 431.9 430.2 424.0 429.3 434.4 424.7 431.0 441.1 -65.4

300 437.7 1 432.5 434.7 432.3 425.5 431.4 437.3 426.2 434.3 445.6 -67.5

(d) Pusan

Return

period

(yr)

Annual maxima method

Joint probability

method

Remark

Gumbel Weibull GEV High

extreme

sea level

Low

extreme

sea level

Max.(min.)

observed LSM CMM PWM LSM CMM PWM LSM CMM PWM

30 169.9 168.2 168.4 169.9 168.5 167.9 169.5 168.2 168.5 176.5 -43.9

174.0

(-41.0)

50 172.4 170.4 170.7 172.3 170.5 169.7 172.0 170.3 170.8 179.2 -45.4

100 175.7 173.4 173.7 175.3 173.1 172.0 175.4 173.2 173.8 183.3 -47.2

200 179.1 176.3 176.7 178.2 175.6 174.1 178.6 176.0 176.9 187.7 -48.9

300 181.0 178.1 178.4 179.9 177.0 175.3 179.5 177.7 178.7 190.5 -49.8

193
(e) Mukho

Return

period

(yr)

Annual maxima method

Joint pre met

)bability

lod

Remark

Gumbel Weibull GEV High

extreme

sea level

Low

extreme

sea level

Max.(min.)

LSM CMM PWM LSM CMM PWM LSM CMM PWM observed

22 77.8 75.0 76.2 76.8 74.3 75.5 76.9 74.4 75.5 87.2 -33.8

77.0

(-29.0)

50 83.3 79.8 81.3 81.0 77.5 79.4 81.6 77.8 79.7 91.1 -35.5

100 88.0 83.7 85.7 84.2 79.9 82.4 85.2 80.4 83.0 93.9 -36.8

200 92.7 87.7 90.0 87.2 82.2 85.2 88.7 82.8 86.2 96.7 -37.8

300 95.4 90.0 92.5 88.9 83.4 86.7 90.6 84.1 88.0 98.1 -38.4

LSM : Least sauare method, CMM : Conventional moment method, PWM : Probability weighted moment method

RETURN PERIOD IN TEARS

2 5 10 20 so 100 300r—

(b)

1000.

0

990.0

e 380. 0 > 970.0

<c 960.0 u

(O

LJ 950.0 X

UJ

cc 9M0.0 X 930.0

920.0

SO.O 80.0 90.0 95.0 98.099.0 99. 7 99.9 CUMULflriVE PROBflBILiTT SCALE

RETURN PERIOD IN TEARS

2 S 10 20 SOtOO 300

3rflTISN :

1 INCHEON

1 1 1 1 1 1

LEGEND

1 LSK

- - 1 CHK

1 PMN

1 1 1 t t 1

(c)

1000.0

SO.O SO.O 9S.0 99.0 99.9 CUMULATIVE PR0BABILITT SCALE

RETURN PERIOD IN TERRS

2 5 10 20 S0100300

-1—I—

r

SrnT[SN I INCHEOH

SO.O 00 . 0 9 0 . 095 . 0 9 9.0 99 .9 CUMULATIVE PROBABILITY SCALE

Fig. 2 Probability distributions of extreme sea levels at Incheon (a) Gumbel, (b) Weibull, (c) GEV

approximately 5- 10cm higher than those by the

annual maxima method at every port. This is clear from the fact that the surges are autocorrelated, i.e., successive hourly samples of the

time series are not mutually independent.^^''

^^°' Surges persist for more than one hour. Another

reason is that large surges tend to not occur with extreme tide levels and so the probability

of an extreme total level due to a combination

of extreme tide and extreme surge is lower than

in that the case of their independence.

When using the same method for estimating

the parameters as in the annual maxima method,

the estimated extreme sea levels in Gumbel

distribution tend to be higher than those of other

distributions, and the estimated extreme sea

levels by the least square method tend to be

higher than those of the other methods of

estimating parameters. In the case of annual

maxima method, the extreme sea levels

computed by the least square method in the

Gumbel distribution are the highest of all, and

the CMM and PWM methods of the Weibull

distribution tend to get lower values. In evaluating of the extreme sea levels these

relative deviations shown in Table 6 vary

depending on not only what kinds of

distributions are taken but also how the

parameters are estimated.

The smallest extreme sea level for a certain return period is less than that from the method

giving the highest value corresponding to one-third of the return period. For instance, the

smallest value( 989.5) for 300 years return period

at Incheon is comparable to the value

corresponding to 1(X) years or less of return

period. Therefore to obtain the extreme sea

levels which are essential for planning or

194
designing a coastal structure, it seems desirable

to get the extreme sea levels, by several

methods. One of the three statistical values

(maximum, minimum and mean) of the estimated

extreme sea levels can be taken as a design sea

level for a return period considering the

functional and economical aspects of the coastal

structures.

5. Conclusions

Estimation methods of extreme sea level with data of relatively short term compared to structural life time were presented here to provide a design criteria needed in planning or designing coastal structures. The methods were

applied to the tidal data recorded at several

major ports in Korea, and results from each

method were analyzed and compared. The

major conclusions are summarized as follows;

1) The extreme sea levels by the joint

probability method are approximately 5-10cm

higher than those by the annual maxima method.

2) In the case of annual maxima method,

the extreme sea levels are higher in the Gumbel

distribution than those in any other distributions

when the same method for estimating

parameters is employed.

3) For the same distribution, the extreme

sea levels evaluated by the least square method

tend to be higher than those by any other

methods.

4) The estimated smallest extreme sea level

for a certain return period is less than the

estimate based on the method giving the highest

value corresponding to one-third of the return

period.

The extreme sea levels are necessary for planing or designing the large scale coastal

developments such as new airport construction,

artificial island construction, and reclamation in the coastal zone. It is often the case that

long-term sea level record may not be available

for a specific site of possible coastal

development. Even with short-term data the

joint probability method can give a statistically

acceptable extreme sea levels. Even if the

method is generally found to give a slightly

overestimated value, it is acceptable in a conservative sense from the engineering point of

view.

6. References

[I] Lennon, G. W., A frequency investigation of

abnormally high tide levels at certain west coast

ports, Proc. Instn Civ. Engrs, 25(1963), 451-483.

[2] Suthons, C. T., Frequency of occurrence of

abnormally high sea levels on the east and

south coasts of England, Proc. Instn Civ. Engrs,

25(1963), 433-450.

[3] Blackman, D. L. and Graff, J., The analysis

of annual extreme sea levels at certain ports in

southern England, Proc. Instn Civ. Engrs., Part

2, 65(1978), 339-357.

[4] Graff, J., Concerning the recurrence of

abnormal sea levels, Coastal Engng, 2(1979),

177-187.

[5] Graff, J., An investigation of the frequency

distributions of annual sea level maxima at ports

around Great Britain, Estuarine, Cx)astal and

Shelf Science, 12(1981), 389-449.

[6] Graff, J. and Blackman, D. L., Analysis of

maximum sea levels in southern England, Proc. 16th Coastal Engng Cx)nf, Hamburg, ASCE,

931-947.

[7] Shim, J. S., Kim, S. I. and Kang, S. W.,

Comparison of computing methods for extreme

sea levels. Ocean Research, KORDI, Korea,

11(1989), 43-50.

[8] Pugh, D. T. and Vassie, J. M., Extreme sea

levels from tide and surge probability, Proc. 16th

Coastal Engng Conf., Hamburg, ASCE, 911-930.

[9] Pugh, D. T. and Vassie, J. M., Applications

of the joint probability method for extreme sea

level computations, Proc. Instn Civ. Engrs, Part

2, 69(1980), 959-975.

[10] Walden, A. T., Prescott, P. and Webber, N.

B., An alternative approach to the joint

probability method for extreme high sea level computations. Coastal Eng., 6(1982), 71-82.

[II] Gumbel, E. J., Statistics of extremes,

Columbia Univ. Press, New York, 1958.

[12] Weibull, W., A statistical distribution

function of wide applicability, J. Applied Mech.,

18(1951), 293.

[13] Jenkinson, A. F., The frequency distribution

of the annual maximum (minimum) values of

meteorological elements. Quart. J. Roy. Meteor.

Soc., 81(1955), 158-171.

[14] Isaacson, M.de St. Q. and MacKenzie, N. G., Long-term distribution of ocean waves : A

review, J. Waterways, Port, Cx)astal and Ocean

195
Div., ASCE, 107(1981), 93-109.

[15] Greenwood, J. A., Landwehr, J. M., Matalas,

N. C. and Wallis, J. R., Probability weighted

moments. Water Resour. Res., 15(1979),

1049-1054.

[16] Takeuchi, K. and Tsuchiya, K., PWM

solutions to normal, lognormal and Pearson-III

distributions, JSCE, 393(1988), 95-101.

[17] Landwehr, J. M., Matalas, N. C. and WalUs,

J. R., Probability weighted moments compared

with some traditional techniques in estimating

Gumbel parameters and quantities, Water Resour.

Res., 15(1979), 1055-1064.

196
Limit Properties Of Maxima Of Weighted I.I.D.

Random Variables

Tomkins, R.J.

University of Regina, Regina, Saskatchewan, Canada

Let Zr, = maxjaiXi, • • • , CnXn}, 72 > 1, where {a^} is a positive real sequence and

X\.X2,--- form a sequence of independent, identically-distributed random variables. Define Z = lim Zn. It will be shown that P[Z < +oc] = 0 or 1. Necessarv and sufficient n—oc conditions will be given for Z to be finite almost surely, or to be almost surely constant.

This work is a preUminary step in the study of the stability of the sequence {Zn}-

1. Introduction

Throughout this paper, Xi,X2,---, will be a sequence of independent, identically- distributed

(i.i.d.) random variables (r.v.) with common

distribution function (d.f.) F{x). Define

Xq = sup{x : F{x) < 1}; (1-1)

note that xq is well-defined, and Xq < +oc. During the past half-century, a good deal of

attention has been paid to the limiting behaviour

of the sequence of maxima {Mn, n > 1}. where

= max{Xi,---,Xn}. (1.2)

The sequence {Mn} is said to be relatively stable

(respectively, almost surely (a.s.) stable) if a real

sequence exists such that

in probability (resp., a.s.) as n ^ oc. Necessary and sufficient conditions for relative stability and for a.s. stability are well-known: see Gnedenko (Ref. [1]), Barndorff- Nielsen (Ref. [2]), and Resnick and Tomkins (Ref. [3]).

It is natural to wonder if analogous results exist for the case where the X„'s are independent,

but not necessarily identically distributed. As

a first step in achieving such a generalization of

the i.i.d. results, it seems reasonable to focus on

the maximum sequence {Zn,n > 1}, where

Zn = m.ax{aiXi,- .anXn] (1.3)

for some positive real sequence {c„, n > 1}. Obviously. Mn and Zn are one and the same if a„ = 1 for all n. Since Z„ is non-decreasing

in n, it makes sense to define

Z = lim Zn. (1.4) n—oo

The limiting behaviour of Mn is straightforward; it is easy to see that

lim Mn = 2^0 a.s. (1.5) n—oo

In other words, the a.s. limit of the M^^-sequence

is the right-hand end-point of the support of

F{x): thus the stability problem for {Mn} is trivial and of little interest in the case where xq is

197
finite. But, not surprisingly, the behaviour of Z„

is somewhat more complex.

For example, if the Xn-sequence is uniformlydistributed on (0,1), and if an = n,n > 1, then

it follows from the Borel Zero-One Law that, for

every M > 0,

P[nXn > M infinitely often (i.o.)] = 1,

and hence it follows that Z = -\-oo a.s. in this

case, even though xo is finite {xq = 1). On

the other hand, if each Xn is exponentially dis- tributed with mean one and a^, = 1/ log (n + l),

then

P[anXn > 2 i.o.] = 0

by the Borel- Cantelli Lemma, from which it is

clear that Z is a.s. finite in this case, even though

To = -|-0O.

The goal of this paper is to determine completely the properties of Z. It will be shown

that P[\Z\ < +oo] = 0 or 1, whatever the value

of Xq may be. Section 2 will deal with the case where xq = +oo; a criterion for Z to be a.s.

finite in this case will be given, and it will be

shown and that Z cannot be a.s. constant in

this case. Section 3 will present necessary and

sufficient conditions for Z to be a.s. finite, and

for Z to be a.s. constant, in the case where xq is

finite. The paper will conclude with some results

and remarks on stability questions in Section 4.

2. Properties of Z when xq = -foe

Throughout this section, it will be assumed

that Xo = +oo; that is, F{x) < 1 for all real x. Some fundamental properties of Z will now be

presented.

Theorem 2.1. Let Xi,X2,--- be i.i.d. r.v. with d.f. F{x) such that F{x) < 1 for all x. Let

{an} be a positive real sequence, and define Z

by (1.4). Then

(i) Z > 7 a.s. for some real 7 if

53{l-F(7<^)} = +oo; (2.1)

n=l

(ii) Z = -1-00 a.s. if (2.1) holds for every real

7;

(iii) if

00 X;{l-F(7a;i)}<«D (2.2)

n=l

for some 7, then Z is a.s. finite, Z is nondegenerate and P[Z < 7] > 0; and

(iv) If Z is a.s. finite then —* 0 as n 00.

Proof. If (2.1) holds for some 7, then

P[anXn > 7 i.o.] = 1 by the Borel Zero- One

Law. It follows that

P[Z > 7] = P[Zn > 7 i.o.] > P[anXn > 7 i.o.] = 1;

i.e., Z > 7 a.s., establishing (i). Part (ii) foUows

easily from (i). Now suppose that (2.2) holds for some 7. Re00 calling that, for 0 < < 1, n > 1, con- n=l

verges (to a positive number) if and only if (iff) 00 00 ^

(1 - Cn) < 00, it follows that JJ ^(70"^) >

n=l n=l

0, and P[anXn > 7 i.o.] = 0 by the BorelCantelli Lemma. Hence, it makes sense to define

the r.v. N = N~, a.s follows:

= <

max{n : a^Xn > 7}

if CnXn > 7 for some n . (2.3)

0 if anXn < 7 for all n > 1

Consequently,

P[Z < 7] = P[N = 0] = P[anXn < 7 for all n > 1] 00 = n ^(^On') > 0. n=l

198
On the other hand,

P[Z > 7] = P[N > 1] > P[aiXi > 7] = 1-F(7ari)>0,

so Z is not degenerate. Moreover, Z = Zk on

the event [A' = ^ > 1, and Z < 7 on [.V = 0], so Z is a.s. finite. This proves (iii).

Finally, if Z is a.s. finite, it foUows from (ii) that (2.2) holds for some real 7. But then, of necessity, F{fa~^) ^ 1 as n oc. Since xq = +00

by hypothesis, it follows that 0, proving

part (iv).

Remarks. 1. It is an easy consequence

of Theorem 2.1 that P[Z = +oc] is zero or one. This is no surprise, since [Z = +oc] =

[ lim max(av-X',v, .anXn) = +oc] for everv n—'oo N > 1, so that [Z = -foe] is a tail event

and, hence, has probability zero or one by Kolmogorov's Zero-One Law.

2. It is evident from Theorem 2.1 that Z is

a.s. finite iff (2.2) holds for some 7 > 0.

.3. The converse to part (iv) is not generally

true. For instance, if F{x) = 1 — ,x > 0, and

Cn = (loglog(n -1- 2))"-^ for n > 1, then a-n ^ 0

and Xq = +0C. But the series in (2.2) equals

cc (log n)~"', which diverges for all 7. Hence

n=3

Z = -hoc a.s. by Theorem 2.1 (ii),

4. In the special case where {an} is a monotone sequence, parts (i) and (ii) of Theorem 2.1 can be derived with the aid of a theorem of Mucci

(Ref. [4], or see Theorem 4.4.1 of Galambos's

book (Ref. [5])).

Corollary 2.2. Let an,Xn, Z and F be as given in Theorem 2.1. Define 70 to be the infi- mum of aU 7, if any, such that (2.2) holds and let

7o = +00 if (2.1) holds for all 7. Then Z > 70

a.s. and P[Z < 70 + c] > 0 for every c > 0.

Proof. If 70 = -f 00 then Z = -l-oo = 70 by

Theorem 2.1 (ii); the second part is trivial in

this case. If 70 < 00 then Z > fo — £ a.s. for

every £ > 0 by Theorem 2.1 (i); hence Z > 70 a.s. An application of part (iii) of the same theorem

concludes the proof.

Remark. 5. If 70 < 00, it follows from Corollary 2.2 and the proof of Theorem 2.1 (iii) that,

for any s > 0, lo<Z<{fo + s)IiN = 0) + ZNliN > 1),

where N = N-^^+s is defined by (2.3).

3. Properties of Z w^hen xq is finite ' Throughout this section, it will be assumed

that xq < +oc: that is, that Xi is a.s. bounded

above. A simple criterion for Z to be a bona

fide r.v. when xq > 0 will now be presented

in Theorem 3.1, which also provides a necessary

and sufficient condition for Z to be a.s. constant.

Theorem 3.1. Let Xi,X2,--- be a sequence

of i.i.d. r.v. with d.f. F, and let {a„} be a

positive sequence. Define xq and Z by (1.1) and

(1.4) respectively, and assume 0 < xq < -Hoc. Then

(i) Z = -Hoc a.s. iff lim sup — +00; n—'OO

(ii) Z < +00 a.s. iff lim sup < cxd; in which

n—'OO case Z < xosupcTi; and

n>l

(iii) Z is a.s. constant iff sup a-n < oc n>l

and either Xi is degenerate or lim sup a-n =

n—'CO supa^j. In either case, Z = xosupajj.

n>l n>l

Proof. Since xq > 0 by hypothesis, P[Xi >

0] > 0. Therefore, > 0 i.o.] = 1 by the Borel Zero- One Law, for every subsequence {rik}. Put another way, this says that

P[Xn, = X+ i.o.] = 1, where X+ = max(X,0),

as usual.

Suppose lim sup a„ = -foe. Then, for any

n— 00 given M > 0, a subsequence {rik} exists such

199
that > M for A; > 1. Hence

Z > lim ma.x(arL,Xn,) = lim max(a„^X^)

k —-co i<.k k—'OC i<k ' > lim Minax(Xni,--- ,XnJ

= Mxq a.s.,

in view of (1.5). Clearly, then, Z = +00 a.s. in

this case. Now suppose lim sup Gn < 00 and define (3 =

n—*oo sup On- Then /3 < 00 and

n>l

Zn < = m.ax(aiX^)

i<n < xq max(ai ,

• • •

, ) < l3xo a.s.

for each n > 1. Therefore, Z < 3xo a.s. Thus,

(i) and (ii) are established.

Now, suppose Z — \ a.s. for some constant

A. Then, from (ii), Q < oc and, as shown above,

A < /3xo. Moreover, A > 0 since P\Z > 0] >

P\X\ > 0] > 0, in view of the assumption xq >

0. But if A < /Sxq, then am > A/xq for some m > 1. Hence, if 7 satisfies A < 7 < 07^2^0,

P[Z > 7] > P[amXm >l] = l- Fiia^^) > 0

by (1.1), since 70^^ < xq. This contradicts the

assumption that Z — \ a.s., so A = /3xo.

Now assume that Xi is not degenerate and

lim sup On < sup On . Then h = sup < /3 n—'oo n>l n>N

for some A'"; clearly N > 2. Define Z/v.n =

max (aiXi) for n > N. Since P[Xn > 0 i.o.] =

N<i<n

1,

Y/v = lim Z^n — lim max (a^X^)

n —00 ' n-*<yoJ\J<i<n

< b lim < bxo < (5xq a.s.

n—*(x> where {Mn] is defined by (1.2). But Z„ =

max(Z7v-i, .^,v,n) for n > TV, so taking n ^ 00

yields Z = max(Z,v-i, X/v) ^-S- But YXr < I^xq

and Z = [3xq, so Za'_i = Z = Pxq a.s. Thus

Ziv-i > 0 a.s., so

/3xo = ^iv-i < /3Miv-i < /3a;o

in view of (1.1) and (1.2). Hence Mn-i = xq

a.s. Consequently,

1 = P[Mn-i = xo] = 1 - (P[Xi < xo])^-^

It follows that P[Xi < xo] = 0 and therefore, by

(1.1), P[Xi = Xq] = 1; that is, Xi is degenerate, a contradiction. Hence, either lim sup an = P 01

n—*oo Xi is degenerate if Z = A a.s.

Finally, turn to the converse. Assume /3 < oc.

If Xi is degenerate, then P[Xi = xq] = 1. Then

Zn = Xo max(ai, • • • , Cn), so that Z = /3xo; i.e., Z is a.s. constant.

Now suppose sup = P for all > 1. Then,

n>N

for every £ > 0, a sequence {n^} exists such that

Un^, > /3 - £,k > 1. Note that Zn < l3Mn if Zn > 0. But P[Xn^ > 0 i.o.] = 1, so Z > 0 and,

by (1.4) and (1.5),

/3xo - lim PMn > Z > lim max(an,X^ ) n—00 fc —oc i<k ' > (/5 - £) lim maxX^ = iP - ^)xo a.s.

It follows that Z = l3xo a.s.

It remains to consider the behaviour of the

Zn-sequence when xo < 0. Theorem 3.2. Let {a„} be a positive real

sequence, and let Xi, X2, • • • be i.i.d. r.v. Define

Xo and Z by (1.1) and (1.4) respectively, and

assume xq < 0. Then — oc < Z < 0 a.s. In fact,

(i) Z is not degenerate iff either (a) xq = 0 and

00 E{l-^(-7«n)}<^ (3.1)

n=l

for some 7 > 0; or (b) xq < 0 and

liminf an > inf On. n—foo n>l

(ii) if neither (a) nor (b) holds, then

Z = xq inf dn a.s. (3.2)

n>l

200
Proof. The proof will be accomplished by considering three cases. Ceise 1: xq = 0 and P[Xi = 0] > 0. By the

Borel Zero-One Law, P[Xn = 0 i.o.] = 1. But

Zn < 0 since xq = 0, so P[Zn = 0 i.o.] = 1 inasmuch as = 0 if Xn = 0 in this case. It follows that Z = 0 a.s. and (3.2) holds.

For the remainder of the proof, it can now

be assumed that P[Xi < 0] = 1, so that X* =

— is a bona fide, positive r.v. for n > 1. Define a^^ = and = max(a^Xi, • • • ,a*X]!^).

It is easy to use (1.3) to check that

Zn = -l/Z: (3.3)

and

Z = -1/Z* if Z" < +0C a.s., (3.4)

where Z" = lim Z".

n^oc

Case 2. xq = O.P[Xi = 0] = 0. Note that X*

is unbounded on the right since xq — 0, so that

{a^} and {X^} obey the conditions of Theorem

2.1. It is readily seen that (3.1) is equivalent to

Y,PK>aa-^]<cc, (3.5) n =l where q = 'f~^ . By Theorem 2.1 (iii), if (3.1)

holds, then Z' is a.s. positive, non-degenerate

and finite. In view of (3.4), Z is a.s. negative,

non-degenerate and finite.

If (3.1) does not hold for any 7, then the series

in (3.5) diverges for aU q and, hence, Z* = +oc

a.s. by Theorem 2.1 (ii). It follows from (3.3)

that Z = 0 a.s. in this case, and hence (3.2)

holds.

Case 3: Xq < 0. In this case, the a.s. least

upper bound on X^ is -Xq'^, so Theorem 3.1 is

pertinent. Suppose

liminf = inf fln- (3-6)

n—oc n>l

This is true if liminf On = 0, in which case

n—'oo lim sup = -l-oo so that Z' = -i-00 a.s. by

n—»oo

Theorem 3.2 (i). Hence Z = 0 a.s. by (3.3), and

(3.2) clearly holds.

If liminf Cn > 0 then lim sup a* < 00, and

n—00 hence Z* - and, by (3.4), Z - is a.s. finite and

non-zero, in view of Theorem 3.1(ii). If (3.6)

also holds, then lim sup a* = sup a*, so that Z*

n—00 n>l

is degenerate by Theorem 3.1 (iii). In fact, Z* =

— Xq^ sup a* a.s., so (3.2) holds, in view of (3.4).

n>l

It remains only to note that, by Theorem 3.1,

Z* is a.s. finite, non-zero and non-degenerate if liminf > 0 and (3.6) is false; consequently Z

n—'oo has the same properties, by (3.4).

4. Connections with stability

The first result of this section links the limiting

behaviour of Zn and MnTheorem 4.1. Let Xi, X2, • • be an i.i.d. se- quence with d.f. F{x) with F{x) < 1 for aU x. Let {cn} be a non-increasing sequence such that an —' 0. Define M„ and Z as in (1.2) and (1.4)

respectively, and 70 as in Corollary 2.2. Then

lim sup CnMn = A a.s. (4.1) n—'OO

for some real A iff

P[Z > A] = 1 and

(4.2)

P[Z < A + £] > 0 for every e>0

and for some A. Moreover, A = 70 in either case.

Proof. It is well-known that

P[Mn > bn i.o.] = P[Xn > bn l.o]

if {bn} is non-decreasing and bn 00. By the

Borel Zero-One Law, then, (4.1) holds iff

00 J^i^-Fifa-')} (4.3)

converges or diverges according as 7 > A or 7 <

A. Hence A = 70 by definition of the latter, and

(4.2) holds with A = 70 by Corollary 2.2.

201
Now assume (4.2) holds. Since F\Z < A+1] >

0,Z < +(X) a.s. by Theorem 2.1. Therefore, the

series in (4.3) converges for some 7, so that 70

is finite. It follows that (4.3) converges if 7 > 70

and diverges if 7 < 70. As noted above, this is tantamount to lim sup CnMn = 70 a.s. n—CO

By considering some special values for {an},

a new necessary and sufficient condition for the

a.s. stability of {M^} arises.

Corollary 4.2. Let {X^}, {M^} and F be as

in Theorem 4.1. Define, for n > 1, = F~^{\ —

n"-^) and Z' = lim max{Xi/^i, • • • , X^/^nli n —'Oo where F~^{x) = inf{y : F(y) > x}. Then {Mn}

is a.s. stable iff P[Z' < 1 + ^] > 0 for every £ > 0.

References

[1] Gnedenko, B.V., Sur la distribution limite

du terme maximum d'une series aleatoire, Ann.

Math., 44(1943), 423-453.

[2] Barndorff-Nielsen, 0., On the limit behaviour of extreme order statistics, Ann. Math.

Statist., 34(1963), 992-1002.

[3] Resnick, S.I. and Tomkins, R.J., Almost

sure stability of maxima, J. Appl. Probab.,

10(1973), 387-401.

[4] Mucci, R., Limit Theorems for Extremes,

Ph.D. Thesis, Temple University, 1977.

[5] Galambos, J., The Asymptotic Theory of

Extreme Order Statistics, 2nd ed., Robert E.

Krieger, Malabar, 1987.

Proof. By Theorem 1 of Resnick and Tomkins

(Ref. [3]), the a.s. stability of {M„} is equivalent

to hm sup Mn//in = 1 a.s. Since it is readily n—<-oo apparent that fin < /^n+i for n > 1 and ^ 00,

taking = l^n^,n > 1, in Theorem 4.1 reveals

that {Mn} is a.s. stable iff P[Z' > 1] = 1 and

P[Z' < 1 + £] > 0 for every £ > 0. But, by

definition of 1 — F{'^^n) > for all n > 1 and 7 < 1, so 70 > 1. If P[Z' < 1 + £] > 0 for

every £ > 0 then, by Theorem 2.1 and Corollary

2.2, 70 is finite and Z' > 70 a.s. If 70 > 1 then

the contradiction P[Z' < 1 -|- £] = 0 arises for

£ < 70 - 1. Hence 70 = 1 and P[Z' > 1] = 1 when P[Z' < 1 -f £] > 0 for every £ > 0. The

result is now apparent.

An obvious open question relates to the stability of the sequence {Z^}. Some work in this vein

has been done by Mucci (Ref. [4], or see The- orems 4.4.1 and 4.4.2 of Ref. [5]). The author

plans to explore this topic in a future paper.

202
Large Deviations For Order Statistics

Vinogradov, V.

Concordia University, Montreal, Quebec, Canada

The presence of two polar types of the formation of large deviations (rare events) is reviewed from the point of view of the extreme value theory. Special consideration is given to the study of the asymptotic behavior of maxima for typical representatives of both polar types:

normal samples and samples with regularly varying tails. The tail approximation/extreme value

approximation alternative is suggested for the case of the normal sample. The results are compared

with those obtained by P. Hall, R.L. Smith and J.P. Cohen. We also derive limit theorems on

large deviations for trimmed sums and pose a number of open problems.

Let {X„ , n > 1} be i.i.d. random variables with

common distribution function F('); denote the

corresponding order statistics by X„ (n) <...< X, (n), and X, + ...+ X„ by S„ ; Sq :=0. Set aAb := min(a,b). In this work, we study probabilities of large

deviationsfor order statistics and their sums, i.e., the

asymptotics of the probabilities such as P{X, (n) > y}, P{X, (n) > y},

P{S„ -X, (n) Xk (n) > y}, etc., where n, y and k

vary such that these probabilities tend to zero. Note

that our definition of large deviations (rare events)

provides a more general approach than the large

deviation principle, as well as the approach to large

deviations as some refinements of theorems on weak

convergence or laws of large numbers.

Let us first consider the case when the random

sample {X^ , n > 1} has the standard normal

distribution. It is well known that in this case the

distribution of the properly centered and normalized

maximum X,(n) converges weakly to the Gumbel

(double exponential) distribution

A(x) := exp{ -e " }:

(1)

P{X, (n) < A, + B„-x} A(x)

as n -> 00, where

A„:=(2-log n)"^-l/2<log log n+log(4n))<2 -log n)-"^

and Bp := (2 -log n)'"^. Note that (1) remains true if A,, and B„ are replaced by a„ and b„ respectively, such that b„/B„ -> 1 and (a^ - A, )/B„ -> 0 as n ^ oo (see, e.g., Ref. [1] Lemma 2.2.2). It was knovra since the works

Ref. [2] and Ref. [3] (cf. also Ref [4]) that the rate of

convergence in (1) is very slow and worst on the tails. Let us now quote the rigorous result. It was obtained

in Ref. [5] that the exact bounds in (1) are as follows:

There exists a positive constant C such that for any

integer n > 2,

3^ < sup I P{X, (n) < A'„ + B'„ - A(x) I X e

(1')

log n

'

where

B'„ := (A'J'; AV= (2-log n)'^^ - (log log n

log(47i)) <8-log n)-^'^ - [(log log n + \og{A%)f -

4<log log n + log(4;i)] • (8<2- log n)''' )"'. The following representation for the distribution of

the maximum from the normal sample containing the

leading error term for (1 ') can be derived from formula

(10) of Ref. [5] and Theorem 2 of Ref. [6]:

203
P{X,(n)< A'„ + BVx} - A(x)

(2)

72 log n

as n -» 00 uniformly in x e R'. Note that formula (10) of Ref [5] is in fact an

auxiliary result of that work used for the derivation

of (!'), whereas slightly different centering and

normalizing sequences were chosen in Ref [6]. Let

us also point out that Theorem 2 of Ref [6] covers a wide class of distributions (which contains the

normal distribution) known as class N.

In addition, in order to emphasize the fact that the rate of convergence in ( 1 ) is very slow and worst on the tails, we now quote the following remark

given on p. 492 of Ref. [7]: "... an approximation

by an extreme value distribution is of little use in determining a critical point for the rejection of

outliers. What is needed is a non-uniform estimate

Qn (x) of P{X,(n) <x}. Ideally such an estimate

should be simple to calculate, and the relative error

lQ„(x)-P{X,(n) < x}l / (l-P{X,(n) < x})

should tend to zero as x and n tend to infinity. " In

this respect, the following result was derived in Ref.

[7] Theorem 3, which can be viewed as a non- uniform estimate in (1) taking into account righthand large deviations.

Let a^ > 0 be defined from the equation

27t V-exp{-a„^} = n^

and let

z„(x) := (27i)-''2 • n • x' • exp{-xV2}.

Then for x > a^,

exp{-z„(x) [l-x-^+3x-^ + z„(x)/2(n-l)]}

(3)

<P{X,(n)<x}<exp{-z„(x)-[l-x-^]}.

Now, let us point out that Theorem 3 of Ref [8]

(see also Remark on p. 11 95 therein) implies that the

asymptotics of the probabilities of right-hand large

deviations for the centered and normalized maximum

from the normal sample is given by the tail of

function A only in every narrow range of large

deviations, namely as n —> 00 and x —>• oo, x = o((log n)''^) (4) P{X,(n)>A„ + B„-x} ~ 1- A(x).

In addition, it is easily shown that (3) implies the

following more general result describing the exact asymptotics of the probabilities of right-hand large

deviations in the fiill range:

P{X,(n)>a, + b„-x}

(4')

~ (1 -AU)) . exp{-xV2ai}

as n CO, X = x(n) oo, where b^ := a„"', and a^ is

the same as in (3); a„~ (21og n )"^ as n -> oo. Remark. It is obvious that in the range of deviations

X = o((log n)''^), the asymptotics of P{Xi(n) > a„+ b„-

x} is completely determined by the first factor

(compare to (4)). On the other hand, for

X > Const ' (log n)''^ x = o(log n),

the asymptotics of P{X,(n) > \+\- x} is completely

determined by the product

(1- A(x)) • exp{-xV2a,^},

whereas for x > Const • log n, all the three factors on

the right-hand side of (4') should be taken into account.

In this work, we develop an alternative method

(hereinafter referred to as the tail approximation),

which provides asymptotic expansions for the

probabilities of the right-hand large deviations along

with accurate estimates of remainders. The just

mentioned method is based on the following apparent

representation, which is equally applicable when the

distribution fimction F of i.i.d. random variables

{X„, n > 1 } is arbitrary (We do not even require the

distribution fimction of properly centered and

normalized maximum X,(n) to belong to the domain of

attraction of anyone limiting distribution):

P{X,(n) > y}

(5)

•PGi^ : X,^> y X,>yi

204
It is obvious that (5) easily follows from the apparent

representation

P ix^in) >y} = Pidix^ > y} ) i=l and the fact that the latter probability can be easily

rewritten by means of the well known formula for

the probability of a union of non-disjoint events. In particular, (5) implies that

P{X,(n) > y}

(5') = n • (1 - F(y)) + 0((n • (1- F(y)f)

as n 00, y -> 00, such that n • (1- F(y)) 0. An application of Representation (5) to the

normal sample yields the following result: Theorem 1. Let us assume that the i.i.d. sequence

{X„, n > 1 } is the standard normal, and denote their

common Laplace distribution function by O. Let a^ be defined as in (3), and bn := a^"'. Then for any

positive X, P{X,(n) > a, + b„ X}

(6)

exp{-x2/2a^l •{la^- (1+x/a^)2n2

l-3-...-(2i-l) ^...}pc.

Proof of Theorem 1 is straightforward. It involves an application of (5) with y = a„+ b^- x, an

expansion of l-O(x) over powers of x as x -> oo (see, e.g., Ref. [9] (vol. I, Chapter 7, Section 7, Problem 1 ) and the fact that

Remarks, (i) Note that both alternating sums on the

right-hand side of (6) can be dropped at any term;

the absolute value of the emerged error will be

bounded by the absolute value of the first omitted

term.

In particular. Theorem 1 implies that

P{X.(n)>a„ + b„-x} =

(6')

(l-A(x))

exp{-x2/2an

}

1+x/aJ

{ 1- aj-(l+x/a^)2

+ 0 \ (1-AU))2 _ exp{-xV2a^}

j 1+x/a^

+ 0

as n

( u-Au)).^^Ei:£!Zifll)

a^(l+x/a„^)5

00, X 00.

(ii) Note that our representations (6) and (6') are similar to Theorem 3 of Ref. [7]. However, our results seem to be more convenient for computations. In

addition, the proposed tail approximation method is equally applicable to an arbitrary distribution function,

whereas the range of applications of Theorem 3 of Ref

[7] is confined to the normal samples. In this respect,

let us quote the following remark from Ref. [10] (see

p. 329 therein): "Hall (1980) suggested

approximations to ^"(x) using some refined inequalities

for the normal tailfimction. These approximations are much closer to <^"(x) than the penultimate

approximations. Thus, if the X^'s are indeed

independent and identically normally distributed and if n is known, then Hall (1980) gives better estimates of

the distribution of

Y„ = max{Xj} than approximations based on extreme

value theory. However, in practice we are often

imcertain of the normality, the independence and

perhaps the value of n. Since the three limit laws

apply to a large class ofinitial distributions, and often

in the dependent case (cf. Galambos (1978)), extreme

value theory approximations are more robust than the

alternatives suggested by Hall (1980).

Now, in view of the above remark, let us suggest

the following approach, which in our opinion provides

a more appropriate approximation for distributions of

205
as

theory.

the tail

jjlications of this A Cc^iiiiucJ vy die uormal sample.

of

Making simple transformations and neglecting the

second order terms we get that the above inequalities are fulfilled for a > xV(2a„^) + x/ia,').

Hence, we obtain the result that for fixed a and, n the

relative error of the extreme value approximation of

P{X,(n) > a„ + b„ • x} is less than e if

n -F^- n Hence,

'{X,(n)>a,+ b„-x}

1.

ill^Li li., ill the relative error of the

+b„*x} by the leftmos

of (6') is less than a) if

imed to be small

(X,(n)>

and side

X,(n,a) := ^ .{(i..4-iog4ii)^/^ -l}

,

accurate at least for the valij rficiently sma ipared to (log n)' . Moreover,

repi on (6') implies that

P in) > a^+b^ Aix))

X < X^in, e) := (1+2 • v

It is not surprising that for any positive

(sufficiently small) e, and for any integer n> N(£):=[(log(^l)/£))^ /8£] + 1, there is an overlap between a lower bound Xi(n, e) and

an upper bound X2(n, e). In other words, for n > N(£) we can always provide an approximation for the distribution of the maximum from the normal

sample whose relative error is less than ( a given a

priori) £. Moreover, there exists a certain range of

deviations in which both the tail approximation and the

extreme value approximation give comparable results. Let us summarize the above discussion m the

following algorithm: assume that we should find an

approximation ofP{X,(n)>a„+b„vc} for given n and x, whose relative error would be less than s. Than we

can choose the tail approximation if X >Xj(n, e) or the extreme value approximation if x <

X2(n, e). This is possible at leastfor n > N(e).

Open Problems, (i) To extend the suggested

alternative to a wider class of distributions (for

example, class N introduced in Ref. [6]).

(ii) To derive sharper estimates for ranges of

applications of both approximations. In particular, it would be interesting to find the border function X (n,£)

such that for X < X (n,£) the extreme value

approximation should be chosen, whereas for

X > X (n,£) the tail approximation should be chosen.

(iii) To derive the asymptotics of the left-hand large

deviations.

Now, let us proceed with the consideration of the

asymptotics of the distribution of the maximum Xi(n)

from the sample of i.i.d. random variables whose

common distribution function F(*) has the right-hand

tail of power type:

(7)

1- F(x) = c„,-x-^' + o(x-"')

as X -» oo, where c„, > 0 and ttj > 0. Note that it was

proved in Ref. [11] Theorem 4 that under fulfilment of

206
(7), the properly normalized distribution function of

the maximum Xi(n),

F„(x):=P{X,(n)<(c,/n)''--x},

converges weakly as n ^ oo to the limiting

distribution T„,(0 defined as "F^^x) := exp{-x"'} if X > 0, and

*P„, := 0 otherwise. It is also well known (cf., e.g., Ref. [12]) that under fulfihnent of (7) and certain

supplementary conditions the following results on the

asymptotic behavior of the probabilities of large

deviations are valid:

P{S„>y}~n-c„,-y-~P{X,(n)>y}

(8)

~l-^a, ((y/(c..-n)

as n 00 where > Y(n) is a positive monotone

sequence that depends on certain parameters and

increases to infinity as n -> oo. Note that the

rightmost relationship in (8) remains true even as n

00, y/n ^'"'-> 00, and just under fulfilment of (7), without

any supplementary restriction (cf., e.g., Section 2.3 of Ref. [13]).

Thus, it is clear in view of (8) and the above

mentioned result on weak convergence to •) that the tail approximation and the extreme value

approximation actually coincide in this case, and we

do not have any alternative but the extreme value

approximation.

Now, let us proceed with the derivation of

refinements to (8). It seems reasonable to assume,

that if more precise information on the tail behavior

of function F is available (compare to (7)), then more precise representations for P{S„ > y} and

P{X,(n) > y} as n -> oo than those, valid up to equivalence can be derived. Various expansions for

P{S„ > y} refining the first of relationships (8) have

been constructed in a number of works by the author

(cf., e.g., Ref. [14,15]) in the case, where the right

hand tail of F admits the following expansion over

negative powers of x:

1- F(x)

(9)

as X 00, where c„, >0, and

0 < a, < ttj < ... < ttj < r.

On the other hand, the following result, Theorem

2, provides the asymptotic expansions for the

distribution of Xi(n) refining both the above mentioned

result on weak convergence of F„ towards <I>„, obtained

in Ref. [1 1] and the rightmost relationship in (8) in the case in which condition (9) is valid. The remainder

term of the expansion of this theorem may be viewed

as the unremovable error generated by the lack of the

perfect information on the tail behavior of F. Theorem 2 (cf. Ref. [16] Theorem 1). Let Condition

(9) be fulfilled. Then

F„(x) = P{X,(n)<(c„,-n)'^"-x}

E

in-y ^ • (V —

[(r/jo-s-«2)/«i]Vo

where 6(x,n)-»0 as n-^oo unifoniily on the rays [C,

+00);

;=l,and| : = ( -1) ^-s- (s+i) . .

• (s+ic-l) /ir!

207
for integer k > 1 (here C > 0 being fixed).

Remarks, (i) Note that a uniform version of our Theorem 2 for a special case f = 2, a2 < 2 - a,,

r = ttj was obtained in Ref. [17] (see Example 1 in Section 6 therein).

(ii) Note that one can easily obtain the asymptotic

expansions for distributions of the maximum X,(n) in the case, where

t l-FU) = 5] c..- (a-x)"^+ o( (a-x)^)

i-l

as x —> a, where a < co, c„i > 0, and 0 < a, < a2 <

... < a, < r by reformulating the result of Theorem

2. Note that distributions of such type often arise in various statistical estimation problems (cf , e.g., Ref.

[18] for details).

Now, let us emphasize that the comparison of the

above results related to maxima from normal samples

with those related to maxima from samples of

distributions having right-hand tails of the power

type reveals the presence of two polar types of the

limiting behavior of the probabilities of large

deviations. Note that the presence of these two polar

types has been first established within the framework

of the classical scheme of summation of independent

identically distributed random variables. Thereupon,

it also became apparent during the study of the

limiting behavior of the probabilities of large

deviations of certain families of stochastic processes

(cf., e.g., Ref [19] Introduction). Following Ref.

[19], we regard the first type of the limiting

behavior of the probabilities of large deviations as one, being associated with the case of fulfilment of

Cramer's condition of the finiteness of the

exponential moment:

E{z'Xj } < 00 for any z e R\ In this case, the

probability of a large deviation is generated mainly

by approximately equal individual summands X^. In

contrast to that, the second polar type is characterized with the case in which the main part of

the probability of a large deviation is generated by

one large summand comparable with the whole sum

S„. The typical example is the case of power tails with index a^<2 (cf condition (7) and relationships

(8) above). Let us point out that only the polar types

of the limiting behavior of the probabilities of large

deviations do not cover all possible cases. Thus, few

subtle results intermediate between the polar types are also known (see, e.g., Ref. [20-22] ). The latter two

papers contain a number of subtle results on the exact asymptotics of (conditional and unconditional)

probablilites of large deviations for trimmed sums

under fialfilment of condition (7) and its left-hand

analog, obtained by fransferring the problem to the

cadlag space D[0,1]. Those results on the asymptotic

behavior of the probabilities of large deviations of

trimmed sums can be interesting from the point of

view of possible applications. On the other hand, they

also provide a better understanding of the nature of

large deviations. To explain this, we introduce the

random step-ftmction

S„,y(t) := S[„,,/yiftG [0,1].

We consider the realizations of S^y (•) in D[0,1]

equipped with the uniform mefric p. The question

arises is what are the typical paths of S^y like if an

event of small probability (a large deviation) has

occurred. It is well known that if the random stepfiinction (r.s.f) S„y is constructed starting from

random variables with finite exponential moments, then

a large deviation is mainly contributed by close-tocontinuous (or even close-to-smooth) paths (cf. Ref

[19] Introduction) - the result of the first polar type.

On the other hand, under fiilfilment of condition (7)

and its left-hand analog with index

a, < 2, large deviations of S „y occur mainly via

almost piecewise constant paths, which perform one or

several big jumps - the result of the second polar type.

For the case of ftilfilment of condition (7) and its left- hand analog with index a, > 2 (note that in this case

the weak convergence of S „y„ (*) to the Wiener

process w( •) holds) the paths of both types can give

comparable contributions to the probability of a large

deviation - the result which is intermediate between the

both polar types). Let us emphasize that this is possible for the case a, > 2 and in a very narrow range

of large deviations only. Outside this range of

deviations, if y > n"^""', events of small probability occur mainly due to the almost piecewise constant

paths (here k > 0 being any real). Moreover, for several sets of D[0,1] large deviations occur via the

paths which perform one or several big jumps and

close-to-continuous functions between them.

Surprisingly, all the discovered sets with such

properties are related to trimmed sums. Now, let us state the following theorem (refining

Theorem 2 of Ref [21] for a special case), which

contains the asymptotic expansion for the probabilities

of the right-hand large deviations of

208
Sn-X,(n): (10)

Theorem 3. Let condition (9) and the left-hand

analog of condition (7) be fulfilled with

a, 6 (0,1) u (1,2).

LetEXi = 0 if a, e (1,2). Then

P{S„ - X,(n) > y}

isisjsl:

as n ^ 00, y / n"°' —> oo. Proof of Theorem 3 is straightforward and purely

probabilistic. It follows along the same lines as that of Ref [21] Theorem 2 and Ref [22], Theorem 1, in which the results on the asymptotic behavior (up to equivalence) of the probabilities of large deviations

of trimmed sums have been obtained. Applying

slight modifications of auxiliary results of those

works, we get that there exists k e (0, 1/2) such that

P{S„- X,(n) > y}

= E > y, \X^\> Ky, |;f^|>Ky}

as n -> CO, y / n'^"' oo. On the other hand, it is not difficult to show that the sum over i and j on the right-hand side of this representation is equal to

<1-2K) y [2]' ! P^n-Mo>y-z^ ^B^.^^)

It is obvious, in view of (9), that the right-hand tail of

the distribution of X„., a X „ has the following

asymptotics as v -> 00: P{X„.,aX„>v}

(11)

Splitting the integral in (10) into three parts by analogy

to the proof of Theorem 1 of Ref [23] (see (7) - (1 1) therein) and replacing P{X„., a X,, > y-z} by the sum

over i and j on the right-hand side of (11) imply the

result of Theorem 3. Remark. Note that from our perspective, the study of

the asymptotic behavior of trimmed sums in the case

in which the distributions of maxima belong to the

domain of attraction of the Gumbel distribution is not

of the same interest. Indeed, it is essentially similar to the study of the asymptotic behavior of S„ (cf , e.g., Ref. [24] for the results on weak convergence, and also

the discussion below Theorem 2 of the present work

given from the point of view of the presence of the

two polar types of the limiting behavior of probabilities

of large deviations).

Open Problems, (i) It seems possible to construct more accurate asymptotic expansions for P{S„ - X,(n) > y} compare to those of Theorem 3.

(ii) To extend Theorem 3 to the case when an

arbifrary fixed number of upper order statistics are deleted from S„.

(iii) To extend Theorem 3 to the case of power tails (cf condition (7)) with index a, > 2.

209
' ^ ^at The prepai-atic-^ ~ ' z

p. : ,;ion of this work ^''^^ . supported by an NSERC Cana-j lational

Research osted at Carleton University,

Ottawa. The ^x. thanks Donald A. i3awson and

Miklos Csorgo for their advice during the preparation

of this work, as well as their hospitality. The

technical assistance of Yaacov B.L. Shapiro, during

the preparation of the final version is ve»3' much

appreciated.

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into account large deviations in the limit theorem on

non-normal convergence to the normal law, C. R.

Math. Rep. Acad. Sci. Canada, 14 (1992), 285-290.

[16] Vinogradov, V., Limit theorems for extreme

order statistics: large deviations and asymptotic

expansions with non-uniform estimates of remainders,

C. R. Math. Rep. Acad. Sci. Canada, 14 (1992), 131-

136.

[17] Smith, R.L., Uniform rates of convergence in extreme-value theory. Adv. Appl. Probab., 14 (1982),

600-622.

[18] Ibragiraov, LA. and Has'minskii, R.Z., Statistical Estimation - Asymptotic Theoiy, Springer, New York,

1981.

[19] Wentzell, A.D., Limit Theorems on Large

Deviations for Markov Stochastic Processes, Kluwer,

Dordredcht, 1990.

[20] Pinelis, LF., A problem of large deviations in a

. r , Qf trajectories, Theory Probab. Appl., 26 (1981),

[21] Godovan'chuk, V.V. and Vinogradov, V., Large

deviations of sums of independent random variables

V +^out several maximal summands. Theory Probab.

34 (1989), 512-515.

[22] Godovan'chuk, V.V. and Vinogradov, V., On

large deviations for trimmed sums of independent

random variables. In: Probability Theory and

Mathematical Statistics. Proc. 5* Vilnius Conf., Vol. 1 (eds.: B. Grigelionis et al.), VSP/Mokslas,

UtrechtA^ilnius, 1990, pp. 424-432.

[23] Vinogradov, V., Asymptotic expansions of the

probability of large deviations for sums of independent

random variables under violation of Cramer's

condition, Theor. Probability and Math. Statist., 31

(1985), 21-27.

[24] Lo, G.S., A note on the asymptotic normality of sums of extreme values, J. Statist. Planning Inference,

22 (1989), 127-136.

210
Extremes Fc

Weissman, L

Technion-Israel Institute of Technolog}'-, Haifa, Israel

Extreme value theory for nonstationary sequences of independent random variables

is discussed. We present limit distributions for extremes, point processes associated with

extremes, extremal processes, record values and record times. The results shown are those

which, we think, are useful for practitioners and are not found in textbooks. The last two

sections include some new results.

1 Introduction

The literature of extreme value theory and its applications is huge and continues to grow. Articles on the subject appear frequently in journals of al- most aJl sciences. A great deal of the literature is devoted to sequences of independent, identically

distributed (iid) random variables (rv's). The in- terested practitioner can find the necessary material in texts such as Refs. [1-5].

A common generalization of the iid sequence is the stationary sequence. Reference [6] provides an

extensive coverage of extreme value theory for sta- tionary sequences. Another generalization is the

independent nonstationary sequences. Except for

two sections in Ref. [2], this case is not discussed

in texts. Dependent nonstationary sequences are discussed in this volume in Ref. [7]. The purpose

of the present paper is to bring to the attention

of practitioners of all sciences the kinds of results

available in the literature. The last two sections

contain some new results.

2 Limiting Distributions for

Extremes

Let {Xi} be a sequence of independent rv's, where

Fi is the distribution function (df) of X,-, and let

M-ak = ^^-th largest of {Xi, X2, . . .., Xn} • The df of Mn = Mni is given by

n P{Mn <x} = l[ Fiix) = Hn{x) . (2.1)

1=1 What are the possible limits, as n 00, of

The answer is trivial, since every df G can appear

as a limit of (2.1) (just take Fi = ' ). In order to avoid trivialities, we normalize Mn with constants

a„ > 0 and 6„ such that as n —>• 00

P{Mn < anX -f hn] = (2.2)

n = \[ Fi{anX + bn)

i=l = 'Hn{cinX -f hn) -> G{x)

at all continuity points of G (assumed to be nondegenerate) and such that

lim min Fi{anX -|- hn) — I {x> xi) . (2.3) n—*co Ki<n

211
Here xl = xl{G) = sup{x : G{x) = 0}. Condition (2.3) is called the uniform right-negligibility

(URN) condition. Under URN, no finite set of Xi

can play a predominant role in determining the

maximum Mn as n oo.

For a df G let \{x) = Xg{x) = -log6'(a:). Define the following classes of df 's: — {G : X{x) is convex}

A^"*" = {G : XL > —oo and X{xl + e^) is convex} = {G : XR < oo and A(a;/? — e~^) is convex}.

Here xr = xr{G) = inf{x : G{x) = 1}. The characterization of the possible limit laws of M„ is given by the following theorem.

Theorem 2.1 (Refs. [8,9]) A df G can be a limit

in (2.2) under (2.3) if and only if it belongs to M = m°\jm^\jm- {=m°\jm-) .

Note that if \{xi-\-e^) is convex so is A(x) (-co <

x < oo), thus M'^ C Af". The classical extreme

value distributions (EVD) A(x) = exp{ — e~^},

^a{^) = exp{-a:-"} {x > 0) and *a(a;) =

exp{— |x|"} {x < 0) belong respectively to M°,M+ and Af". The df F{x) = x"" {a > 0, 0 <

X < 1) belongs to each one of the three subclasses.

The normal distribution is not in M. So, in situations where the iid assumption is not justified, the

practitioner might fit a df for the extremes from

a much larger class than the EVD.

Note that if G G A^, then G is strictly increasing, continuous and diflferentiable inside {xl,xr).

The right-end xr can be a point of discontinuity

only if G € M~.

Note that under (2.3), Hn(an+i^ + ^n+i) a-nd Hn+i{cLn+ix + bn+i) have the same limit (as n

oo), thus the Convergence of Types Theorem (Ref.

[10], p. 253) impHes

1 , —> 0 (n ^ oo) .

(2.4)

If A(x) is convex, then G(x)/G{x + A) is a df for

every A > 0. Thus, for every increasing sequence

{bn} with bo = Q, bn oo which satisfied (2.4),

the df's Fi{x) = G{x - 6,)/G(x - 6,_i) satisfy

(2.2) and (2.3) (with an = 1). If G € M+, then

G{xL + x)/G{xL + ocx) is a df for a > 1 and thus

for every increasing sequence {a„} (a^ > 0, a„ ^

oo) which satisfies (2.4), the df's Fi{x) = G{xl -f- x/ai)/G(xi:-fx/a,_i) satisfy (2.2) and (2.3) (with

bn = -dn^L)- Finally, if G G Af~, then G{xr -\- x)/G{xR -f ax) is a df for 0 < a < 1 and thus for every decreasing sequence {a^} (a^ > 0, Cn

0), which satisfies (2.4), the df's Fi{x) = G{xr +

x/a,)/G(xfi-l-x/a,_i) satisfy (2.2) and (2.3) (with

K = -o-uXr).

Let Xni = {Xi — bn)/an and let J„ be the point

process of the points {Xni i = l,...,n}; that

is Jn{x, oo) = ^"=1 1{Xni > x) IS the number of

exceed ances in the sample over the level a^x +

bn. Since P{I{Xni > x) = 1} = 1 - Fi{anX +

bn) = Fni(x), under the URN condition, (2.2) is equivalent to

n Fniix) —> - log G(x) = A(x) (n oo) .

(2.5)

Let TUni = {Mni — bn)/an', then the points {nini -

i = l,...,n} are the points of Jn in descending

order. We have the following Poisson convergence.

Theorem 2.2 (Ref. [11]) Under (2.2) and (2.3),

there exists a nonhomogeneous Poisson process

J on {xl,xr) whose mean measure at (x,oo) is A(x). Moreover, if mi > m2 > • • - are the points

of J in descending order, then for each k

{mni,...,mnk) ^ imi,...,mk) {n oo) . (2.6)

Notice that {A(m,)} are the points of a standard

homogeneous Poisson process (SHPP) on (0,oo);

thus a simple exercise shows that the joint density

of (mi, . .

. , mk) is

k i;{xi,...,Xk) = G{xk)ll{-X'ixi))

i=l

(xi > X2 > •••> Xfc), (2.7)

where A'(x) is the derivative of A(x). The

marginal df of m^- is given by

212
lim P{mnk < 2r} = (2.8) n—oo

k-l = P{mk < a;} = ^(a:) A'(x)/i!

= / e-^'u^-^dulik-iy. . J\{x)

The results (2.4)-(2.8) are the same as in the iid case, except that G belongs to A<, a much larger

class than the classical EVD.

Suppose a df G is a candidate for the limiting df

of Afn, then for each fixed k and large n we should

have

{\{{Mni-hr,)lan):i=l.....k}

S{r.:i = l,...,A:}

where {T,} axe the points of an SHPP.

Suppose n is large. Fix a /c «C n\ thus if G is the right df then

{Mni .i= 1,.

§ {anA"^(r,) + 6n : i = 1,...,/:} .

A graphical method to verify that G is indeed

the right df is to plot Mni vs. (we replace

Ti by its expectation ETi = i) for i =

If the points are scattered around a straight Une,

then we have statistical evidence in favor of this G.

Moreover, the slope and intercept are estimates

of and bn, respectively. Another possible approach is maximum likelihood. Suppose one wants

to fit £ as the limiting df for Adri, where 0 is a parameter to be estimated. Then the Ukelihood

based on Mni » • • • 5 ^nk is approximately

L{e,an,bn) = a-^Ge{{Mnk

k -hn)lan)'Y[{-X'e{{M,n - bn)/an)) • t=l

The triple {d,an-,bn) which maximizes L will be

used as our estimates. This approach is similar to

Ref. [12] or Ref. [13], except that Ge is not limited

to the EVD class. For the iid case, let Ciip) a-nd ^n{p) be the p- quantiles of Xi and Mn, respectively. Then for aJI 0 < p < 1 Up) = 6(p'/") ~ 6(1 + (log p)/n) . (2.9)

The classical ixtreme value theory implies

{Up)-bn)/an-^aP) = G-\p) (n-oo).

(2.10)

In the general case, where X,- ~ Fi, (2.9) is not

necessarily true but (2.10) still holds.

3 Functional Limit Theorems

Suppose one is interested not just in the maximum

Mn of the sample, but also in its evolution along

time. The process {M[nt] : ^ > 0} (Mq = Xj) is a pure-jump Markov process, whose distinct values

consist of the set of upper records of the sequence

{X,}; M[n.] jumps at t if and only if Xnt is an

upper record. Let m„i(i) = {M[nt] — bn)/o,ni then

we have

[nt] P{mni{t) <x} = ll Fi{anX + bn) = (3.1)

J=l / anX + {bn - b[nt])

, , \ = H^n.^^a^nt] — ^hnt^j .

Suppose HnidnX + bn) G{x) {G being nondegenerate). Then (3.1) will have a nondegenerate

limit if and only if ^ at ; — Pt {n^ oo) (3.2)

for some constants at > 0 and (3f Moreover, in

this case limn-+oo -P{"^ni(0 ^ x} = G{atX + (3t) =

Gt{x). In fact, we have the following result.

Theorem 3.1 Suppose (2.2) and (2.3) hold.

Then

lim P{mni{t) <x} = Gt{x) (3.3) n—CO= G{atX + Pt) (OO)

if and only if (3.2) holds for all t > 0. Moreover,

if {at,(3t) is not identically (1, 0), (3.3) implies the

URN condition (2.3).

213
Note that in the iid case F"(a„.T + bn) —>• implies both the URN condition F{anX + bn) 1 and F["*J(a„.T + G\x) for aU f > 0 (i.e. (3.3)). Reference [14] shows tha-t cc^,/?; must have

the form

A = c(i " tn/p (3,4)

for some constants p and c (interpret (1 — t'')//? a.s

,0 = 0). The convergence of the rn.arginaJ.s

oi 'it,-,].! ) implies afunctional Umit theorem.

Theorem 3,2 (Ref. [15]) Under (3,3), ilieri eo>

ists a vvr^ -ivniv Mnr^-^v rrocess mi = {mi{t) :

t . . / 0 < i'l < • < ik and

Xy < X2 < " - < Xi;

P{mni{ti) < Xi : i = 1, . . ., P{mi{t;) < Xi :

h.{^i)/Gt,^,{x,)] {Goir:)^l) (3.5)

J=1.

Theorem. 3.2 claims that aJl the finite dimensional laws (f<'''' ' '

' '1 verge to those of mi

(write m„i , j i _ - nat Tnni{ ) is right con- tinuous and we ran always choose a right contin- uous version ' •). Therefore, both m„i and

mi a.re elem/ ' Skorohod space .0(0, co)

with the Ji-'* v leference [16] proves full weak convergent' ^co)-

The process m^ is called an extj^emal process Bxid

its transition probabilities for i,s > 0 are given by

Gt{y)

0

mit) = x}

y > a:

y < x .

For joint convergence of ^ = {'^'^[nt]i ~

bn)/o>n {i > 1), jt is useful i iploy the point

process Kn of the points {{i/n., X^,-)}. Let K be a Poisson point process on IR-f. x IR, vi/hose mean

measure a.t (0,t] x (a:,oo) is Xt{x) — -]ogGt{x).

If {(Tj,l',)} 3.re the points of ii, let mk{t) be the

/cth largest Y{ among points with T,- < t. Then

obviously we have

P{mk{t) < x} =

= P{K{{Q, t] X (a;, oo)) < - 1}

==Gtix)y2K{^)/i'--

t=0

The m_ain result of this section is the following:

Theorem.

point ?;

3fs. [16, 17]) Under (3.3), the

:\n converges to the nonhomogeya process K. Moreover, for each k,

D

{vini,. . .,mnk) (mi,. . .,mk) in D''{0,oo)

4 T

J

Records

Given a sequence of random variables {X,- : i >

1), Xj is an (upper) record if Xj > Mj_i; when

all Fi are continuous, Xj is a record if Xj = Mj.

The indices {L{j) : j > 1} (X(l) = 1), where the

Markov process {Mk ' k > 1} jumps are called

record times and the values {Mi^j^ : j > 1} =

{^^L{j) • J ^ 1} ^-re the record values. Let Nn be

the number of records among the first n observations.

V dues and record times for iid sequences

and tiieiT relation to Poisson and the extremal process are treated extensively in Ref. [3]. A lovely

review of this subject is provided by Ref. [18]. Records of independent nonstationary sequences

are treated in Refs. [19-25]. Reference [21] is motivated by the unpredicted high sea levels in The

Netherlands that caused the collapse of the sea

dikes and the loss of 2000 human lives; aU the

other authors are motivated by sport-records. The

.models treated can be presented as follows. The

sequence of independent rv's {Xj : j > 1} is such

tha.t

X, = Zj+cd, (i=l,2,...), (4.1)

vvhere c is a constant, {dj} is a monotone se- quence, dj 6 IR, and {Zj} are iid with a common

214
df F. Here the df of Xj is Fj{x) = F{x - cdj),

i.e. aJl the Fj are of the same type. Climatologists

who believe in global warming can use these models for their data. For a linear-growth model we

have the following useful result.

Theorem 4.1 (Ref. [19]) Suppose EZ^ <

DO, c > 0 and dj = j.

(a) There exists p 6 [0,1] such that as n ~* oo Nn/n —> p a.s. and in and L{n)/n —

a.s.

(b) If F is continuous, E{X^y < oo and 0 <

p < 1 then as n CO

V^{n-H{n) -p~^)^ A/'(0,p-V)

for some a"^ — cr^(p).

The record rate p is obtained from

/CO

Jl F{x + cj)dF{x) , -oo^ • . J=l

(4.2)

where pk = F{Ak} = P{Xk is a record}. Note

that ENn = YliPj- Ballerini and Resnick show

that Ak and Ak+m tend to be independent for

k,m large. When F{x) = A((x - b)/a) (a > 0),

the {Aj} are mutually independent, pj = 6j/Sj,

where Sj = exj>{cj/a) and Sj = Yl^-i^i (Smith

and MiUer (1984)) and

p=l- e-^/'^ . (4.3)

Let A„ = L{n + 1) — L{n) be the inter-record

times.

Theorem 4.2 (Ref. [20]) For F{x) = K{{x -

b)/a), c > 0 and dj = j, as n oo we have

{An+k : > 1} S {r,- : > 1} , (4.4)

where {Tk} are iid geometric with p = 1 — e~'^l°-

The two results Nnfn —> p a.s. and (4.4) are complementary; by Theorem 4.1 pj — P{Aj} —

p. Thus, the independence of the Aj implies that

for 8.11 k = k{n) —> oo and j > 1, the number of records among {A'jt-fi, - . • , Xjt+j} tends to Bin{j,p). Hence, the inter- record times must con- verge to independent geometric rv's with parameter p.

Rf: :all thai 'hsn c — 0 (•-.'.•. when the Xj are iid)

then Pj = 1 ENn iogn, the Aj are independent and Nr,/ log n ~> 1 a.s. iioreover, EAk = oo

for all k > 1 and (log A/;)//: 1 a.s. {k oo).

Ballerini and Resnick analyze the mile-race data

in Ref. [19j. The fastest time Xj of every year

(1860-1982) is plotted vs. j {j = 1,...,123) and

indeed ;

' ; exhibited. There are 36

records ii . : , si the record-rates {Nj/j : j = 1, 2, . .

. , 123} do stabilize around p = 36/123

from about j - 50 to the end.

Let us assume now that F £ ^'(A), i.e. F'^[anX -i- bn) A(x) fox some constants > 0

and 6n. Define the extrema^ '"or{Zj}and

{Xj = Zj+cbj} by

ml{t)= f Z[^^]} -bn)lan

and

m^{i) r= (Mfni] - {1 + C)br,)lan .

arly mf ---^ m.c in .D(0, oo), where is an

extremal process as in (3.5) with Gtix) = A'(x) =

A(a; - log t). We have the folJov/ing result for .

Theorem 4,3 (Ref. [21]) For c> 0, as n oo

vie in .D(0, oo)

where

{m,{t) : t > 0} = {mo{t'-^'/ic+ 1)) : t > 0} . (4.5)

The properties of can be read off easily from

the properties of nio via (4,5). In particular, for

large n and y, P{iV4 < y} ^ F"'^' f^^+'^y). It means that behaves like a maximum of n'^'^^

iid jw's (whose common df is jr^/Cc+i)), This implies an analogous result to Theorem 4.1(a).

215
Theorem 4.4 (Ref. [21]) For Nn, the number of

records among {Xj = Zj + cbj : j — I, . . . ,n}

with c> 0, we have

(C + 1) log 71 1 a.s. and in L2 . (4-6)

Notice that when F = A, we have 6n =

logn, Cn = 1 and Xj = Zj + c logj, i.e. — we

have a logarithmic growth.

Another result of a similar nature is the following.

Theorem 4.5 (Ref. [21]) Let {Zj} be iid with

F{x) = $i(x) = e-^/^(a: > 0). Let Nn be the number of records among {Xj = Zj+j : j = 1, . . . , n}.

Then

Nn

log^ n

Var

Nr^ log^ n

Note, here we have a linear-growth model, but

since Zi > 0,EZi = 00, Theorem 4.1 does not

apply.

5 Extreme Value Times

Let Rnk = min{i < n : Mjk = Mnk}, so Rni =

L{Nn) is the last record time in {l,...,n}, Rn2

is the time (or index) of the second largest observation, etc. Asymptotic results for Rni are taken

from Ref. [26]. Results for Rnk (k > 1) are new.

Observe first that

P{Rni <k, Mn< x] = P{M{k,n) < Mk < x}

(5.1)

= /. n Fi{y)dllF,{y) , i=k+i j=i

where M{k,n) = max{A7-+i, . . .,Xn}. Thus, we

have the foDowing asymptotic result.

Theorem 5.1 (Ref. [26]) Under (3.3) we have

forO<t<l

lim P{Rni < nt, mni < x} = (5.2) n—00 J \ / G{y) - L

dGtiy)

-00 Gt{y)

G{y)d\ogGt{y) = Hi{t,x).

Let Hi{t) = Hi{t,oo) be the limiting df of

then (3.3) is only sufficient for the existence of a

limiting distribution for Rni (take Fi = F, where

F is not in any domain of attraction of an EVD,

but Rni/n C/(0, 1) (uniform on [0,1])).

Theorem 5.1 is generalized as follows.

Theorem 5.2 Under (3.3) we have fork > 1 and

0 <t <l

lim P{Rnk < n't, mnk ^ x} = (5.3)

- 1: ^^y\'-\y)dGt{y)^Hk{t,x)

Gt{y)

and for 0 < 5, t <\

lim P{Rni < nt, Rn2 < ns} = (5.4)

G{x)

Xt{x)dGs{x) = Qit,s).

'-00 Gs{x)

For the special case

Gt(x) = G'^^'\x) (5.5)

for some function 4> we have the following results.

Theorem 5.3 Under (3.3) and (5.5) we have

(i) (p{t) = f for some 7 > 0 and G must be an EVD.

(ii) Hk{t) = Hk{t, 00) = <j>(t) {k>l)

(Hi) For each k, Rnk cind Mnk ore (asymptotically) independent.

(iv) For each k, Rni,..., Rnk ore (asymptotically) independent.

We have a 0-1 law.

Theorem 5.4 Under (3.3), if for some k (fixed)

P

P-nk/n Ck, where Ck is a constant, then Ck = c and c = 0 or 1.

216
6 Two Growth Models

Let {Zj : j > 1} be iid, Zj ~ A and let Xj = Zj + dj {dj € IR). Define 6j = exp(c?j) and

Sn = Ei^j- Then Fjix) = A(x - dj) = A^^(x)

and P{Mn < x} = A^"{x). Hence the right nor- malization is Cn = 1, bn = log5n and

P{mn{t) <x} = P{M[^t] -bn<x]

= A^(-'l/^"(x) {t>0).

Since Fj(x + 6„) = A*>/^"(2), URN holds when

6, = oiSn) (1 < i < n).

(i) Logarithmic models. Suppose dj = c logj.

For URN we need c > —1. If c > -1, we have

bn = log(n'=+V(c+ 1)) and P{mn{t) < x}

Gt{x) = A''^\x) = A(x-(c+l)logO. Thus (3.2)

holds with (Qt,/3t) = (1, (c+ l)logO. By Theorem

5.3, Hkit) = t^+^ (0 < t < 1, k > 1). Moreover,

the extreme vsJues and their times of occurrence

are (sLsymptotically) independent. The de HaanVerkade result Nn (c + l)logn (a.s.) holds in

fact for c > — 1 and not just for c > 0.

For c = -1 we have bn = log log n, (3.2) holds

with (at,/5f) = (1,0), which means that m{i) =

m(l) a.s. {t > 0) — the extremes occur very early.

Indeed, here, since Gt = G, Hk{t) = 1 (i > 0)

(i.e. Rnh/TT' 0) and 7V„ ~ loglogn (a.s.). For

c < -1, we have 5n 5 < oo and P{Mn < x} =

A'^"(a;) —» A-^(x). URN does not hold, but we can show that P{Rni = k} k^S (k = 1,2,...),

i.e. the Izist record occurs at a finite time w.p.l.

Since iV„ < Rni, Nn remains finite as n ^ oo. This is not surprising since the sequence {Xj} is stochastically decreasing, so the first few observations determine the sample maximum.

(ii) Polynomial models. Let dj = j^. When a = 0

the {Xj} are iid. When a < 0, 6j 1. Thus, the

{Xj} are "aJmost" iid and they yield the same

asymptotic results as the iid sequence.

For a > 0 we have bn = log^n ~ n'* + (1 —

a)logn and bn — b[nt] Hence, m„(<) —> — oo

(a.s.) {0 < t < 1) but P{mn(l) < x} = A(x).

This means, Gt{x) = 1 and Hk{t) = 0 (0 < i < 1),

i.e. Rnk/n 1. As long as 0 < a < 1, URN

holds, pj = 6j/Sj 0, Ya Pj log5n and Nn ~

log5„ (a.s.), i.e. Nn = Oin"). Since P{Rni <

k} = Sk/Sn, one can show that for 0 < a < 1

lim P{(n - Rni)an°-'^ > x} = e'^" (x > 0) . n—CO

(6.1)

For a = 1 (the linear model), URN does not hold, = 6,1 S, = e^-i(e-l)/(e^-l) ->p= 1-e-i -

the case treated by Ballerini and Resnick. Here, ~ np (a.s.) and the inter-record times are (asymptotically) geometric (result (6.1) is actually a generalization of this fact).

When a > l,pj — Sj/Sj 1 very fast, so not

only Nn ~ n but except for finitely many, all Xj

are records.

Acknow^ledgement. This research was supported by the Fund for the Promotion of Research

at the Technion.

References

[1] Gumbel, E.J., Statistics of Extremes,

Columbia University Press, New York (1958).

[2] Galambos, J., The Asymptotic Theory of

Extreme Order Statistics, John Wiley, New

York (1978). (Second edition, Robert E. Krieger, Malabar, Florida (1987)).

[3] Resnick, S.I., Extreme Values, Regular Variation and Point Processes, Springer, New York

(1987).

[4] Castilo, E., Extreme Value Theory in Engineering, Academic Press, New York (1987).

[5] Reiss, R.-D., Approximate Distribution of

Order Statistics, Springer-Veriag, N.Y.

(1989).

[6] Leadbetter, M.R., Lindgren, G. and Rootzen,

H., Extremes and Related Properties of

Random Sequences and Processes, SpringerVerlag, Berlin (1983).

217
[7] Husler, J., Extremes: Limit results for (univariate and multivariate) nonstationary se- quences, this volume (1994).

[8] Mejzler, D., On a Problem of B.V. Gredenko,

Ukrain Math, Z 2, 67-184 (1949) (Russian).

[9] Mejzler, D., On the Problem of the Limit Distributions for the Maximal Term of a Variational Series, Lvov. Politechn. Inst., Nauch.

Zap. Ser. Fig.MatK 38 (1956), 90-109 (Russian).

[10] Feller, W., An Introduction to Probability

Theory and Its Applications, Vol. II, Second

edition, John Wiley, New York (1971).

[11] Mejzler, D. and Weissman, I., On Some Results of N.V. Smirnov Concerning Limit Distributions for Variational Series, Annals of

Mathematical Statistics 40 (1969), 480-491.

[12] Weissman, I., Estimation of Parameters and

Large Quantiles Based on the k Largest Observations, Journal of the American Statistical Association 73 (1978), 812-815.

[13] Smith, R.L. and Weissman, I., Maximum

LikeUhood Estimation of the Lower Tail of a Probability Distribution, Journal of the

Royal Statistical Society, Ser. B 47 (1985),

285-298.

[14] VV^eissman, L, On Location and Scale Functions of a Class of Limiting Processes with

Application to Extreme Value Theory, Annals of Probability 3 (1975b), 178-181.

[15] Weissman, L, Extremal Processes Generated

by Independent Nonidentically Distributed

Random Variables, Annals of Probability 3

(1975a), 172-177.

[16] Weissman, I., On Weak Convergence of Extremal Processes, Annals of Probability 4

(1976), 470-474.

[17] Weissman, I., Multivariate Extremal Processes Generated by Independent Nonidentically Distributed Random Variables, Journal

of Applied Probability 12 (1975c), 477-487.

[18] Click, N., Breaking Records and Breaking Boards, The American Mathematical

Monthly SB (1978), 2-26.

[19] Ballerini, R. and Resnick, S., Records From

Improving Populations, Journal of Applied

Probability 22 (1985), 487-502.

[20] Ballerini, R. and Resnick, S. Records in the

Presence of a Linear Trend, Advances in Applied Probability 19 (1987), 801-828.

[21] de Ha-a.n, L. and Verkade, E., On Extreme

Value Theory in the Presence of a Trend,

Journal of Applied Probability 24 (1987), 62-

76.

[22] Smith, R.L., Forecasting Records by Maximum Likehhood, Journal of the American

Statistical Association 83 (1988), 331-338.

[23] Smith, R.L. and Miller, J.E., Predicting Records, Technical Report, Department

of Mathematics, Imperial College, London

(1984).

[24] Smith, R.L. and Miller, J.E., A NonGaussian State Space Model and Applications to the Prediction of Records, Journal

of the Royal Statistical Society, Ser. B 48

(1986), 79-88.

[25] Yang, M.C.K., On the Distribution of the

Inter- Record Times in an Increasing Population, Journal of Applied Probability 12

(1975), 148-154.

[26] de Haan, L. and Weissman, I., The Index

of the Outstanding Observation Among n

Independent Ones, Stochastic Processes and

Their Applications 27 (1988), 317-329.

218
Order Statistics and F

Idei

Wenocur, R.S,

University of Pennsylvania, Philadelphia, PA

An urn model approach to exceedances of order statistics leads to new proofs of combinatorial identities, one of which is Gauss's 2F1 summation formula. The relationships

among Gauss's 2F1 identity, inverse Po'lya distributions, and order statistics emerge as a consequence. We discuss connections among combinatorial methods of proving hyperge-

©metric identities and our probabilistic approach, with emphasis on exceeding the upper

order statistics, in particular the maximum, of a random sample.

1. INTRODUCTION

Proofs of combinatorial identities have a long

history. For a classic proof of Gauss's 2-P'i summation theorem see for example Slater (Ref. [1], pp. 27-28).

Often a proof of an identity using order statis- tics — in particular extreme order statistics — appears in the literature; see for example

Refs. [2, 3, 4]. The purposes of this paper are: to present proofs of combinatorial identities us- ing order statistics, with emphasis on Gauss's

2F1 summation theorem which generalizes the

result in Ref. [3]; to show relationships with

WZ-pairs which are explained in the following

paragraph; and to illustrate how classic results can be proven by various methods which are the

consequences of posing mathematical questions

in different contexts. Many combinatorial identities can now be verified by means of the Wilf-Zeilberger certifica- tion theorems (Ref. [5]) which provide a method

for certifying couples of identities via WZ-poArs.

If two functions F{N,k) and G{N,k), defined

for integer k and nonnegative integer A'^, satisfy

the condition

AnF = AkG (for integers A > 0 and k),

{F,G) is '""^ •'-ir. Under the conditions thai lii integer k, the limit

fk — limM 00 F{N.,k) exists and is finite; for

' > 0, lim/:_^±ooG(A, /c) 0; and

.0 .Ojv>o ^{^^^ ~^-') = 0, we have a couple

f titles

J]jP(A,/i;) - const. (A = 0, 1, 2, ...), (1)

k and

E G{NJz)= V; (/,--F(0,i)).

iV>0 3<k-l

/ er, com.puterized proofs of hypergeometric identities are now possible (see Ref. [6]).

2. ORDER STATISTICS

As a consequence of studying the works of

Galambos, including Ref. [7], Galambos and

Seneta (Ref. [8]), David (Ref. [9]), and Gum_-

bel (Ref. [10]), we analyzed order statistics from

the following point of view, using an urn ro.odel

r r,pTr,r-cb. (see, for example, Refs. [11, 12, 13]).

3, very clear and concise introduction to order stafAstics and extrem.es, see Gal;

book (Ref. [7], pp. 16-1 7).

219
Let Xi , X2 ,

• . X^r be a random sample from an arbitrary real-valued continu- ous distribution. Call Xi, X2, X^v the

previous sample, and consider future trials Xj\j+i , Xm+2 1

• X^+k 5• • • from the same

distribution. With probability 1, the order

statistics < X(2) < ... < ^{N) ^.s- sociated with the previous sample determine

N + 1 disjoint random intervals /i, /2, .../jv+i on the real line, into which any future observations must fall. For any la, <^ =

1, 2, A'^ -t- 1, the conditional probability

P{XN+k+ l € Ia\XN+l,XN+2 - XN+k) is equal to 77^7^^, where r is the number of

Xj^+j^s, j = 1,2,... A:, that fall into /q. Since

the IqS are almost surely disjoint, determining

where S C {1,2, ...A'^ -1-1}, poses no difficulty.

See for example Refs. [11, 13].

Wenocur, Refs. [11,13] ), it must follow that

00 E PiWNJ,m = m + k) = I . (3)

k=0

4. PROBABILISTIC PROOF

OF GAUSS'S 2F1 IDENTITY

Let us now apply results of §2 and §3 to prove

Gauss's 2F1 identity and to show its relationship

to an inverse Polya distribution.

Gauss's 2F1 summation theorem often appears in the form (see for example Ref. [1], pp.

27-28 or Ref. [4])

^ , , , r(c)r(c -a-b)

ioT c > a + b; where 2^1 [<^? b;c; 1] is defined by

3. EXCEEDANCES

Sample until exactly m future trials exceed the

order statistic Xq) where 1 < j < A'^. If ^N,j,m equals the number of future trials until

X'(j) is exceeded m times, then (adopting the

convention that the empty product equals 1, of

course)

P{WNj^rn = m+k) =

{m + k-l)\{N +l-j)---{N + m-j]

;c! (m- 1)! {N +1)---{N + m)

2Fi[a,6; c; 1]

(N + Tn+ 1) {N + Tn + k)

_ {N + m-j)\ m

~ {N + m)\ {N

(m)(m-H)---(m + k-l)(j)---(j + k-l)

k! (N-fm+ l)---(N + m + k) ^

'

Since P(W^;v,N,m = m + k) = 1 (see Johnk^o son and Kotz, Ref. [14], sections 4.4 and 4.5;

= E —{a + k-l)b---{b+k-l)

k^O c{c+ 1) {€ + k - l)k\

see for example Slater (Ref. [1], p. 1) or Ref. [4]. The purpose of this section is to present a probabilistic proof of (4) where a, 6, and c are positive

integers.

Directly from (2) and (3),

2Fi[m,j;N + m+ 1;1]

_ {N + m)\ {N - j)\ _ r{c)r{c-a-b) ~ {N + m-j)\Nl ~ r(c-a)r(c-6) '

where a = b = j, c = N + m + 1, and where

A'-l-m + l >jt-|-mis equivalent to Gauss's

condition c > a + b . Notice that (2) describes an inverse Pdlya distribution; see Ref. [13], Ref. [14], pp.194-200,

and Ref. [15]. Hence, Gauss's sum is related to

this particular type of waiting-time distribution.

A combinatorial argument leads to a different

proof of Gauss's 2^1 identity in Ref. [4]; nevertheless, the relationship between Gauss's 2P1

sum and the inverse Pdlya distribution emerges.

220
Bose-Einstein statistics are related, too; see, for example the now classic texts of Feller, that of

Galambos, and Refs. [11, 13].

5. PROBABILISTIC PROOF

OF A CLASSIC IDENTITY

WITH A TERMINATING SUM

The urn model approach to order statistics pre- sented in §2 is useful for proving various hypergeometric identities. In Ref. [3], a less general

form of Gauss's identity is proven by means of

a similar but more restrictive method. For dif- ferent techniques which are combinatorial in nature, see Refs. [4, 5, 6]. As another example of applying the approach

of §2, let us derive a result which involves a fi- nite sum. Certainly, this is a well-known identity

that appears in Ref. [16] and in textbooks, but

let us derive it again in the present context for at least two reasons: (a) to illustrate the util- ity of methods employed in this paper; (b) to

lead to an identity (6) that has been of interest in industry (Ref. [17]) but proved by longer

means, namely, an induction argument, that ig- nores order statistics. Suppose we consider n

future trials and let be the probability that

exactly k of these n trials have values less than

the smallest of past N values. The fact n

that H 9fc = 1 and direct application of the

urn model presented in §2 lead to the identity

A probabilistic interpretation of (6) involves the

N*'^ order statistic, (that is, the maximum). Divide both sides of Eq. (6) by its right-hand-side.

Thus, we obtain the sum, from /: = 0 to n, of

the probabilities that exactly k of our n future

values fall below the maximum -X^(iv) of our pre- vious sample.

6. RELATIONSHIP TO SOME WZ-PAIRS

Strong connections exist between the urn model

approach to order statistics and the WilfZeilberger certification theorems of Ref. [5] described in §1. Suppose we restrict our attention

to the j^^ upper order statistic. Let m = 1 and consider exceeding the j*^^

largest (that is, X(jv+i-j)' ^^e (A'" -|- 1 — jY^

smallest ) of the first N values observed. Letting

F{N,k) = P{WN,N+i-j,i = A;) for A; = 1,2,

we have the VTZ-pair

F{N,k) = jN\{N + k-

{N -jy.{N + k)l

G{N,k) =

j {l-k)N\{N + k-l-j)\

{N -j + l)\{N + k)\

Suppose m = 1 and consider exceeding the

maximum ^(at) of the first A'^ values observed.

Letting F{N,k) = P{Wn,n,i = k) for k =

1,2,... , we obtain the VFZ-pair

E

k=0

j + k-1

k

N - j + n - k n — k

N + n n

(5)

F{N,k) =

G{N,k) =

{N + k){N ^-k-l)'

1 - k

{N + k){N + k -1)

'

If j = N = p + 1, (5) reduces to a result from

the ancient field of summation calculus:

k=o

J2{k + l)---{k + p)

_ {n + l)---{n + p + l)

(P + 1)

(6)

and potential function

^{N, k) = -N

(N + k-l)'

where, as defined in Ref. [5], Ak^ = F and = G .

221
Although the notation is different, F{N, k) appears in Refs. [3, 11, 12, 13] as a probability depending on N and k , namely, previous sample

size and stopping time respectively. Probabilistic analysis of F{N, k) presented here, employed

in Pk-efs. [3, 11, 12, 13], and described from an- other perspective in Ref. [4], provides a counterpoint to combinatorial methods in Ref. [5] and

Ref. [6].

ACKNOWLEDGMENTS

Many thanks to Janos Galambos, Herbert S.

V/ilf, and Doron Zeilberger for lively discussions

that led to these results. Additional thanks to

Professor Galambos and to the referee for suggestions that improved the presentation .

REFERENCES

[1] Slater, L. J., Generalized Hypergeometric

Functions, Cambridge University Press, Cambridge, 1966.

[2] Kadell, K. W. J., A probabilistic proof of

Ramanujan's iV'i sum, SIAM J. Math. Anal,

18 (1987), 1539-1548.

[3] Wenocur, R. S., Rediscovery and alternate

proof of Gauss's identity, Ann. Discrete Math,

9 (1980), 79-82.

[4] Wenocur, R. S., A probabilistic proof of

Gauss's 2F1 identity, to appear in J. Comhin.

Th. (1994).

[5] Wilf, H. S. and Zeilberger, D., Rational

functions certify combinatorial identities, J. Am.

Math. Soc, 3 (1990), 147 - 158.

[6] Wilf, H. S. and Zeilberger, D., An algorithmic proof theory for hypergeometric (ordinary

and "q'') multisum/integral identities. Invent,

mathem., 108 (1992), 575-633.

[7] Galambos, J., The Asymptotic Theory of Extreme Order Statistics, John Wiley and Sons,

New York, 1978.

[8] Galambos, J. and Seneta, E., Record times,

Proc. Amer. Math. Soc, 50 (1975), 383 - 387.

[9] David, H. A., Order Statistics, John Wiley

and Sons, New York, 1970.

[10] Gumbel, E. J., Statistics of Extremes,

Columbia University Press, New York, 1958.

[11] Wenocur, R. S., Waiting times and return

periods related to order statistics, PhD thesis,

Temple University, Philadelphia, 1979.

[12] Wenocur, R. S., Waiting times and return

periods to exceed the maximum of a previous

sample, Statis. Distrib. in Scientific Work, 6

(1981), 411 - 418.

[13] Wenocur, R. S., Waiting times and re- turn periods related to order statistics: an

application of urn models, Statis. Distrib. in

Scientific Work, 6 (1981), 419 - 434.

[14] Johnson, N. L. and Kotz, S., Urn Models and their Application, John Wiley and Sons,

New York, 1977.

[15] Janardan, K. G. and Patil, G. P., On multivariate modified Pdlya and inverse Polya distributions and their properties.. Annals of the

Institute of Stat. Math., 26 (1974), 271-276.

[16] Galambos, J., Introductory article: Statistical Extremes and Applications, (ed.: J. Tiago

de Oliveira), Reidel, 1984, 13.

[17] Newsletter, Daniel H. Wagner Associates,

Paoli, 1987.

222
An Examination Of The Extremes Of Selected New Zealand Rainfall And RunOff Records Foi

Evidence Of Trend

Withers, C.S. and Silby, W.W.

Institute for Industrial Research and Development, Lower Hutt, New Zealand

This report examines selected annual NZ rain and river flow maxima and minima for evidence of long term trend. The model assumes a linear time trend while the residuals are from the generalised extreme value (GEV) distribution. This is the most popular model

for dealing with extremes, and gives a first-order approximation to the theory of extremes. The iterative estimation technique combines the high-efficiency L-moment method of

Hosking et al. (1985) for the GEV-parameters with the maximum likelihood equation for slope.

Regions chosen were prone to flood or drought. The results were surprising: the evidence

for the presence of a trend in rainfall and runoff was very weak using the least squares

estimate (LSE) of slope but extremely significant when using our mixed MLE/L-moment

method.

It is recommended that the techniques used here be tried on NZ temperature series, as a greenhouse effect is much more likely to show up as a trend in temperature than a trend in

rainfall. To eliminate the possibility of programming error it is also recommended that the method

be reprogrammed from scratch.

1 INTRODUCTION

This paper examines selected annual New

Zealand (NZ) rain and river flow maxima and

minima for evidence of long term trend. The main results, given in §4, regress these series against time, assuming that the residuals are from

the generalised extreme value (GEV) distribution.

This is the most popular model for dealing with

extremes, and gives a first-order approximation

to the theory of extremes. (Ongoing work aims at improving this approximation by considering the

two main families of distributions that arise in theory: distributions with power tails and

distributions with exponential-power tails.)

§2 describes the data series chosen.

The theory developed for estimating the

parameters is outlined in §3. It is based on an

extension and modification of the method of

L-moments (also called probability weighted

moments) given by Hosking et al. (1985) that

uses the meiximum liklihood equation for the

slope iteratively with the L-moment method for the 3 GEV parameters.

Regions chosen were prone to flood and drought.

The evidence for the presence of a trend in rainfall and runoff was very weak using the least squares estimate (LSE) of slope but extremely

significant when using our mixed

MLE/L-moment method. At first sight this is

223
surprising — partly because examining extremes

throws away a lot of information, partly because one might expect the data series the data series to be too short for confirmation of a trend — and

partly because rainfall need not increase linearly as global C02 increases according to global

circulation models as presently developed. In fact

it may decrease in some regions: see Mullan and

Renwick (1990) for New Zealand climate change

from increased C02 as inferred from a global

circulation model, and Salinger et al (1990) for a

scenarios approach to changes in New Zealand

climate. However since Hosking et al (1985)

showed that the L-moment estimates for GEV

parameters were more efficient than the MLEs,

one does expect that the LSE of slope for a model

with GEV residuals would be inefficient compared with our MLE/L-moment method.

It is intended to work out an analytic form for the asymptotic covariance, both to confirm that

it is regular enough for its jackknife estimate to • be consistent, and to cut down on the long

amount of time jackknifing took — typically over one hour per run, using Splus.

We recommend that our iterative estimation

method be started from the LSE for slope rather

than from slope 0, as in seven out of the 18 runs, beginning iterations from slope 0 gave

convergence to a wrong result. Jump scenarios were also tested for in an ad hoc manner: see §4 for numerical results. Detection of a trend in rainfall is a much more

difficult problem than in temperature, and it is recommended that the methods developed as well as refinements in progress be applied to NZ

temperature series. In addition more theoretical

work needs to be done to settle the question of

the efficiency of regression methods in detecting a

trend over methods based purely on analysis of extremes.

2 THE DATA SERIES CHOSEN

RAINFALL SERIES — MAXIMUM DAILY PER ANNUM in mm

Gisborne 876902 1937-87

Masterton 59604 1926-87

Timaru 414201 1881-1985

Palmerston North 53603 1928-87

Arthur's Pass 219510 1957-89

Arthur's Pass 219501 1941-87

RIVER FLOW SERIES - ANNUAL 15 MINUTE FLOOD PEAKS

(cubic metres/second) - MINIMUM AVERAGE OVER 7 DAYS

(litres/second)

Motu 16502 Flood 1960-1987 Minimum 1960-1990

Ruamahanga 29201 Flood 1955-1987 Minimum 1955-1987

Opihi 69618 Flood 1936-1987 Minimum 1965-1986

Opuha 69614 Flood 1936-1987 Minimum 1965-1986

Manawatu 1032560 Flood 1929-1988 Minimum 1972-1989

Waimakariri 66401 Flood 1930-1987 Minimum 1967-1989

The number following each series is the NZ Meteorological Service's code.

3 THEORY: ESTIMATES

FOR THE GEV

DISTRIBUTION WHEN

A TREND IS PRESENT

Much has been written on the theory of extremes

for the case of stationary observations. See for

example the books of Gumbel (1958), Leadbetter

et al (1980), Galambos (1987) and Resnick

(1987). However these books give very little theory for the case when the observations are not

stationary.

One of the few papers dealing with a search for a

trend is Smith (1989): this paper analyses the

224
number of exceedances of a given (high) level based on 15 years of hourly measurements. He

applies techniques to remove short-term

dependency and seasonality in the data. Then he

uses the 3 parameter generalised extreme value

(GEV) model for exceedances, allowing the

parameters to vary from period co period as well

as from year to year. (Here a 'period' is one or two months.) Parameters are estimated by

maximum liklihood. Fit is measured by plotting

the ordered (estimated) residuals against their expected value for GEV exceedances (the

'generalised Pareto distribution') and also by plotting the ordered values transformed by

their estimated distributions against the expected

value of the order statistics from a uniform

distribution. This paper also gives a number of

useful references, such as Busier (1986) on

extremes of nonstationary sequences. Davison and Smith (1990) proceed further with

this approach, allowing the parameters of this model to be given functions of regression

covariates, with the parameters estimated by

maximimum liklihood or least squares. Tawn (1988) and Smith (1986) give a related approach but use the joint distribution of the r largest extremes rather than the threshold

method.

McKerchar and Pearson (1989) looked at 13 New

Zealand continuous water level recordings of more

than 50 years each and found some evidence

(pl5-16) of long term trend. They fitted the

GEV distribution to annual flood peaks to a

great number of New Zealand flood peak series and noted (p33) that the EV 1 distribution (that

is the GEV distribution with ^ = 0) gave an

unsatisfactory fit to many series. The approach taken here is to fit the model

Yi = bti + Xi,l<i<n (3.1)

where Y, = extreme value for ith observation, ti = year of tth observation (typically i) and the zth residual Xi is assumed to come from the GEV

distribution

F{x) = exp{-A(x)}, A(x) = (1 - kz)'/", (3.2) z = {x — ^)/a where a > 0.

The GEV distribution is the limiting distribution

of

(maxj^i Zj — hN)/cr4 as N —> oo,

if a limit exists for some c^r > 0 and when

Zi^ . . ., Zn is a random sample from some

distribution on R. The following examples are from §4 of Withers (1992a).

EXAMPLE 3.1. If

P{Zi > x) (a/x)'' as X —» oo where a, 6 > 0,

then one can take = 0, cyv = aN^I^

k = -6-1 < 0, a = 6-\ i = l.a

EXAMPLE 3.2. If

P{Zi > x) w fz'^e''' as X —> oo

where z = {(x — b)/c}'^ and a, c > 0, then one can take

bN = cNI'"" {1 + a-^N-^{dN2 + fx)) where

N\ = logN, N2 = log Ni, fi = log f and

cjv = ca-'^Nl^''~\ k = 0, a=l, ^ = 0.D

EXAMPLE 3.3. If Zi is bounded above by b and

P{Zi > x) c{b — x)" as X t 6 where a, c > 0,

then one may take 6;v = a, cn = {cN)~^^'',

k = a = a-^ > 0, ^ = -l.D

By the first two examples, if the density of the

underlying variable Z has upper tail falling to

zero as a power law (or exponentially) then for maxj^j Zj, the corresponding GEV distribution

has k < 0 (or k = 0 respectively).

By the third example (with Yj = —Zj) if Yi is bounded below (as for daily rainfall or riverflow)

then for mirijLi Yj the corresponding GEV

distribution has ^ > 0. Without loss of generality we took

to be zero. (That is we replaced U by — t. This

is the same as reparameterising ^.)

We refer to the 4 parameters of the model as

9 = {b,^,a,k)

If k = 0, A(x) — exp{—z), the limit as ^ —* 00, so F is the "EVr distribution.

The range of x is given by 1 — ^2 > 0, that is

X < ^ + a/k if k>0, and x > ^ + a/k if k <0.

If b is known to be 0, Hosking et al. (1985) have

shown by simulation that the L-estimates (also called PWMs) for {k,^,a) are more eflftcient than

the maiximum liklihood estimates (MLEs) for

225
n = 15, 25, 50 and k = -.4, -.2, 0, .2, .4 and are comparable for n = 100. For n = cx), the relative

efficiencies of these estimates at A: = 0 are about

.8 for k, .95 for ^ and .85 for d, but fall to 0 at k = —.5. The relative efficiency of k falls to 0 at ^ = .5 while the other 2 remain high (Figure 4). The L—estimates do not exist if A: < —1; the mean MLEs do not exist if k > 1/3: see p252,254.

(See Phien (1987) for a comparison of estimates

for the case when k is known to be 0.) However no results are available for the case when b is unknown. Tawn and Dixon (1992) use the model (3.1) on extreme sea levels, estimating

6 by the MLE. Our approach is to use the MLE

equation for 6 iteratively with equations (14),

(15) of Hosking et al for the L—estimates of

{k,^,a) bcised on the estimated residuals,

Xi = Yi— Hi . Beginning with 6 = 0 we found

that in all cases except two, 5 iterations gave

accuracy to at least 4 decimal places.

The derivative of the mean log likelihood ratio

w.r.t. h is —g{b)/a where

by • I + 6c/ k = Xn

(their notation for max Xi).

By their (15) this is equivalent to replacing their (13) by

6o + dr(l + k)/k = x„.

This results in their implicit estimate k of (13)

being replaced by the explicit estimate

^ = - log2{l - (26i - 6o)/(x„ - 6o)} (3.4)

where 6o = X, 6i = (n^ — n)~^S"_2(i - l)xj and

^1 ^ ••• < are the ordered values of Xi, ...,Xn- The corresponding estimates of ^ and a are now

given by their (15) with k replaced by k . To summarise: if k > 1, use k . This will decrease

the variances of the estimates with high

probability.

(With exponentially small probability as n

increases one could find ^ < 1 < ^ ; in that case one cannot rely on either estimate of k.)

9{b) = n-'E7^MK^'^'-'^ + ik-mr') (3.3) 4 NUMERICAL RESULTS

where Wi = 1 - kZi for Z,- = {Xi - gib) was plotted against 6 = 0 for the first and last iterations, and in each case found to be

monotonic with a single root. (See the figures at the end of §5 for some examples.) Thus the

method is not complicated by the presence of

multiple roots. (To evaluate the root we used

Newton's method.)

Our sample sizes ranged from 17 to 95 years and

our estimates k from —.45 to .23 with none

significantly different from 0. Only one b was

significantly different from 0. We end this section with a refinement available to the theory of Hosking et al. when Ar > 1 or ^ > 1. (It turned out that in our applications k was less than one in each case so that this refinement was

not needed.)

Hosking et al. noted that MLEs are 'not always

satisfactory' if A: > 1 as the density approaches oo

as max Xi approaches its upper bound

u = ^ + a/k. In situations where the range

depends on the parameters — in this case X < u, the corresponding estimate — in this case u = max Xi, is superefficient, that is has variance

0(n~^) , not just 0(n~^), so that one can reduce

the variances by replacing say the 3rd of Hosking

et al's 3 L-moment equations, that is their (12),

For each series we give the minimum, quartiles,

maximum, and number of years covered (n),

followed by the number of iterations (5 except for

tv/o cases), parameter estimates, their covariance

as estimated by the jackknife method, the

estimate and variance of the quantile estimates

for the GEV distribution fitted. By (8) of

Hosking et al the F—quantile of the GEV is

x{F) = ^ + a{l- {-logF)''}/k = t{0) (4.1)

This is estimated by t{9) and its variance by

v(^,C) where

V {e,C) = tCi,i = dt{e)/d9

and

' C is the jacknife estimate of the covariance C of ^. We then estimate the years t for which

EYt = bt + EX will equal 1.1 and .9 of EY,—

that is, the mean will increase by 10% or drop by

10%, according to the model (3.1). The t for which the EYt changes to a fraction H of EY is given by

bt + EX = H{bi+ EX) (4.2)

that is by t(e) = Hi + b-^(H - 1)EX where

EX = ^-\- aEZ and EZ = k-^ - T{k).

226
We then estimate the standard deviation of t{6) as for the previous example o{t{9).

Next we compute the goodness-of-fit statistic

nJ2(Yi-Yi)y{in-3)var{X)} (4.3)

i=l

where var{X) = {a/k)^{T{l + 2k) - T(k + l)^}.

(This is only an ad hoc statistic as the usual

regularity conditions for its asymptotic

distribution are not satisfied.)

Finally we give the 2-sample t-statistic for testing

that the first and second half of the data series have the same mean, without assuming equal

variances. Under the null hypothesis of no change

in mean, this is asymptotically normal (0,1).

RAINFALL SERIES — MAXIMUM DAILY PER ANNUM

Mean Years

Years available

Gisborne 876902 1937--87 1962 51

Masterton 59604 1926--87 1956.5 62

Timaru 414201 1881--1985 1937.505 95

Palmerston North 53603 1928--87 1957.5 60

Arthur's Pass 219510 1957--89 1973 33

Arthur's Pass 219501 1941--87 1964 47

RIVER FLOW SERIES - ANNUAL 15 MINUTE FLOOD PEAKS

Motu 16502 1960--1987 1973.5 28

Ruamahanga 29201 1955--1987 1971 33

Opihi 69618 1936--1987 1961.5 52

Opuha 69614 1936--1987 1961.5 52

Manawatu 1032560 1929--1988 1958.5 60

Waimakariri 66401 1930--1987 1958.5 58

NOTE 1 For Timaru, data for the years 1883,1887-1895 is not available. RIVER FLOW SERIES - ANNUAL 15 MINUTE MINIMA

Motu 16502 1960--1991 1975.5 32

Ruamahanga 29201 1955--1987 1971 33

Opihi 69618 1965--1986 1975.5 22

Opuha 69614 1965--74,1978-84 1973.97 17 Manawatu 1032560 1972--1989 1980.5 18

Waimakariri 66401 1967--1989 1978 23

NOTE 2 The minima are for the 7 day averages of the 15 minute observations. The data years for

minima and maxima flow are different as the minima flow came from electronic files and the maxima flows from McKerchar and Pearson

(1989) Appendices 1 and 2. U

Before giving the results in detail, we give the

following summary of the estimates of the slopes

b and the extreme value parameters k, and the

^-statistics for testing the difference in means

between the first and second half of each series.

SUMMARY OF ESTIMATES OF b AND k AND THEIR SIGNIFICANCE AND THE

T-TEST

b b/s.d. k k/s.d. t — stat'c RAINFALL MAXIMA

Gisborne 876902 .451 7.77 -.197 -12.79 .67 Masterton 59604 -.085 -4.16 -.179 -16.27 -.04

Timaru 414201 -.096 -13.92 -.044 -3.69 .15 Palmerston North 53603 -.049 -3.29 -.036 3.50 .77

227
Arthur s Pass 219510 1.000 4.85 .118 4.37 1 1 A

1.12 Arthur's Pass 219501 .445 6.39 .082 4.29 1.36 RIVER FLOOD PEAKS

Motu 16502 .339 .69 .229 9.38 .00 Ruamahanga 29201 2.128 4.82 .102 3.81 1.81 Opihi 69618 1.775 5.98 -.400 -27.93 .62 Opuha 69614 -.198 -.90 -.244 -1.59 -.74

Manawatu 1032560 1.342 A AO 2.43 — AO C .025 A A ^ —2.04 — .34 Waimakariri 664014 .519 .74 -.254 -22.28 -1.01

RIVER FLOW MINIMA

Motu 16502 —27.18 — 10.68 -.039 -14.31 -2.02

Ruamahanga 29201 84.42 N/A .877 N/A 2.04

Opihi 69618 -34.97 -12.00 .366 9.48 -1.56

Opuha 69614 -57.69 -5.63 .766 9.17 -1.10

Manawatu 1032560 169.72 N/A .675 N/A -.14

Waimakariri 66401 52.07 .86 .346 11.32 .08 NOTE 3 The value of b/sd(b) when residuals were GEV turned oui to be larger than that for

normal residuals by a factor of 2 to 5. This could

possibly suggest that convergence to normality is far from being approached at the sample sizes considered here. Although a search for a programming error to explain this effect was

fruitless, this possibility cannot be ruled out.

The most striking feature of the analyses was the

reduction in variance of b when moving from the

LSE to our estimate. The result is to make all the b's highly significant — ie non-zero, whereas

using the LSE one would draw the opposite

conclusion — ie that they were not significantly

different from 0. There are several possible explanations: (a) the

LSE of slope is highly inefficient for the model

trend plus GEV residuals; — recall that our

estimate mixes the MLE method with the

L-moment method , and that the latter is more

efficient than the MLE method. The LSE is efficient for normal residuals but not for GEV

residuals — but it seems surpising that it should

be so inefficient. This feature will be checked by

obtaining an analytic form for the asymptotic

variance; (b) the jacknife estimate of variance is too small because the analytic form for the

asymptotic variance is not regular ( ie. not estimable by replacing the distribution of

residuals by their estimated empirical

distribution); — this is the case for quantiles such

as the median, and will be checked by obtaining an analytic form for the asymptotic variance in the coming year using the method of stochastic

expansion of Withers (1987); however if the

asymptotic variance was not regular one would

expect the jackknife estimate of variance to be

too large, not too small; (c) program error; the

program is listed in Appendix B and has been

carefully checked.

The second feature that jumps out is the highly

significant values of the shape parameters, due to

their small standard deviations — as estimated

by the jackknife. This is no doubt closely

connected with the small s.d.s for the slope

estimate.

In this regard we quote Pearson (1992) p 67: "Three-parameter distributions such as the GEV

should not be used to analyse flood frequency at single sites, unless the annual series is at least 30

years long, since sampling errors are much larger

than for two-parameter distributions". With our four-parameter model variability will be even

greater.

A third surprising feature of the data is that the

slope of b is positive (and highly significant) in six cases but negative (and highly significant) in six cases. (This allows for the fact that negative

minima not minima should be fitted to GEV

distributions: these estimated slopes have been

reversed so that a positive slope indicates an

increase.) This is not so surprising on further

thought as different regions are expected to react

differently to global warming — some will get more rain and some less — though globally the

average rainfall will increase. NOTE 4 These results were run beginning our

iterations with b=0 and also beginning with the

228
LSW for b io check that convergence to the same

results occurred. In fad this failed for seven

cases:

1. for Gisbome maximum rainfall starting at b=0 the final estimates were

b = .156 k = -.194 a = 19.712 | = 63.419

2. for Palmerston North maximum rainfall starting at b=0 the final estimates were

b = -.012 k = -.049 d = 12.583 i = 42.773

3. for Motu flood peak starting at b=0 the final estimates were

b = .950 k = .163 a = 77.546 | = 218.559

4- for Opihi flood peak starting at b=0 the final estimates were

b = .161 k = -.447 a = 55.551 1 = 77.337

5. for Opuha flood peak starting at b=0 the final estimates were

b = 1.210 k = -.280 a = 82.063 | = 124.867

6. for Manawatu flood peak starting at b=0 the final estimates were b = —1.749 k =

-.043 a = 468.015 ^ = 1160.312

7. for Waimakariri flood peak starting at b=0

the final estimates were b = 4.904 k =

-0.2800 a = 361.867 i = 1177.764

This is another reason to begin iteration with the LSE rather than 0. The other was that starting

with b=0 gave some residuals so large thai in some cases the method fails unless those outliers are discarded.

NQTE 5 The fact that the t-tests did not pick

up any differences in mean between the first and

second half of each series is consistent with

explanation (a) above that the LSE is much less efficient than our estimate of slope; but it could

also be argued to be consistent with the explanations (b) and (c).

5 Acknowledgement

The computing was carried out by Sarah Harper

and Howard Silby. This was done in S-plus. A

copy of the program is available on request.

6 REFERENCES

Bradford, E (1991) Investigation of some

statistical methods for detecting global warming.

DSIR Physical Science Report, Box 1335

Wellington.

Bradford, E (1991) Further methods used on

lake flows. — Appendix D of (1991) progress

report by Withers and Pearson listed below,

t^radford, E (1992) Some statistical methods for

detecting changes in mean temperature. DSIR

Physical Science Report 45, Box 1335 Wellington.

Bradford, E (1992) South Island lake inflow : possible trend changes and modelling. DSIR

Physical Sciences Report 51. Applied

Mathematics Group, DSIR, Box 1335,

Wellington.

Bradford, E (1992) Multiple regressions of

rainfall at various NZ locations on time, site, sunspot index and Southern Oscillation index. DSIR Physical Sciences Report, Applied

Mathematics Group, DSIR, Box 1335,

Wellington.

Davison, AC and Smith, RL (1990) Models

for exceedances over high thresholds. J.R. Statist. Soc. B, 393-442.

Galambos, J (1987) The asymptotic theory of extreme order statistics, 2nd edition, Krieger,

Melbourne, Florida.

Gumbel, EJ (1958) Statistics of extremes. Columbia University Press, New York.

Hawkins, DM (1977) Testing a sequence of

observations for a shift in location. JASA 72,

180-186.

Hosking, JRM, Wallis, JR, and Wood, EF

(1985) Estimation of the generalized

extreme-value distribution by the method of

probability-weighted moments. Technometrics 27,

251-261.

Leadbetter, MR, Lindgren, G and

Rootzen, H (1980) Extremes and related properties of random sequences and processes. Springer-Verlag, New York.

McKerchar, AI and Pearson, CP (1989)

Flood frequency in New Zealand. Publication 20,

Hydrology Centre, Christchurch. ISSN 0112-1197.

Mullan, AB and Renwick, J A (1990) Climate

change in the New Zealand region inferredJrom

general circulation models. NZ Meteorological

Service report 30.11.90 for the NZ Ministry for the Environment.

229
Pearson, CP (1992) Analysis of floods and low

flows. Chapter 6 of Mosley, M.P. (1992) (ed)

Waters of New Zealand NZ Hydroligical Society,

Wellington.

Phien, HN (1987) A review of methods of

parameter estimation for the extreme value type-l

distribution. Jnl of Hydrology 90, 251-268.

Resnick, S (1987) Extreme values, regular

variation and point processes. Springer, New

York.

Salinger, MJ, MuUan, AB, Porteous, AS,

Reid, SJ, Thompson, CS, Withers, CS,

Coutts, LA, and Fouhy, E (1990) New

Zealand climate extremes: scenarios for 2050 AD.

Report prepared by NZ Meteorological Service

and DSIR Applied Mathematics Group for the

Ministry for the Environment.

Smith, RL (1986) Extreme value theory based on

the r largest annual events. Journal of Hydrology

86, 27-43.

Smith, RL (1989) Extreme value analysis of

environmental time series: an application to trend detection in ground-level ozone. Statistical Science 4, 367-393.

Tawn, JA (1988) An extreme-value theory model

for dependent observations. Journal of Hydrology

101, 227-250.

Tawn, JA and Dixon, MJ (1992) Trends in extreme sea levels p 313-318 of Proceedings 5th

International Meeting on Statistical Climatology,

Toronto, June 1992.

Withers,CS (1987) The bias and skewness of

M-esiimaies in regression. DSIR Applied

Mathematics Technical Report 139. Withers, CS (1990) Extreme rainfalls recognising mixed processes. - Jointly with K.J. A.

Revfeim. DSIR Applied Mathematics Division

Unpublished Report 1990/013.

Withers, CS and Pearson, CP (1991)

Detecting Climate Change in NZ rainfall and

runoff records. DSIR Physical Sciences Report 22. Withers, CS (1991) Moment estimates for

mixtures with common scale. Communications in

Statistics, Theory and Methods 20, 1445-1461-

Withers, CS (1991) Tests and confidence

intervals for the shape parameter of a gamma

distribution. Communications in Statistics,

Theory and Methods, 20 (7), 2663-2685.

Withers, CS (1992a) Asymptotic multivariate

distributions and moments of extremes, to be

submitted.

Withers, CS (1992b) Expansions for quantiles

and multivariate moments of extremes for a class of Pareto-type distributions. Submitted.

Withers, CS (1992c) Expansions for quantiles

and moments of extremes for distributions of

exponential-power type. Submitted.

Withers, CS (1992d) Expansions for the distribution of the maximum from a Pareto-type

distribution when a trend is present. Submitted.

Withers, CS (1992e) Saddlepoint expansions

and Bell polynomials Submitted.

Withers, CS (1992f) Expansions for the beta function and its inverse for one parameter large.

Submitted.

Withers, CS (1992g) Estimates for mixtures of

compound Poisson-gamma distributions. Submitted.

Withers, CS (1992h) Multivariate Bell

polynomials, series, chain rules, moments and

inversion. Submitted.

Withers, CS (19921) Moment estimates for

mixtures of several distributions with different means or scales. Submitted.

Withers, CS (1992j) The moments and

cumulants of a mixture. Submitted.

Withers, CS (1992k) Estimating trend: Is daily,

monthly or annual data best? Submitted.

Withers, CS (19921) The distribution of the range of a Wiener process. Submitted.

230
Extreme Values In Business Interruption

Insurance

Zajdenweber, D.

Universite de Paris X-Nanterre, Nanterre Cedex, France

The size-distribution of yearly claims in the French business interruption insurance branch is a Pareto law with an extremely long tail. The behavior of that law reflects the fact that the total value of yearly claims is dominated by a small number of major claims. We estimate the

characteristic exponent of the tail, which is very close to one. This value means that the theoretical probability distribution has no expectation, and that business interruption insurance may be a very hazardeous economic activity.

INTRODUCTION.

The insurance industry lies on a

foundation stone : the subdivision of risks through the law of large numbers. More

precisely, in case of a statistical distribution of

claims with a finite expectation, the law of large

numbers proves that the average amount of claim

per head (or per policy) becomes closer and

closer to the expectation as the number of

policies becomes larger. In the business

interruption branch, as in some other branches

where the risks may be catastrophic, Ref [1], the

rate of convergence of the average claim towards

the expectation can be very small. Even worse, it can be nil, because of a small number of

extrranely large claims. In that case, the main

problems posed to managers of insurance

companies are the estimation of the actuarial value of the risks and the estunation of the appropriate amount ofreserves necessary to cope

with tiie extreme values of some claims. In the present paper, we shall only analyze the yearly

size-distribution of business interruption claims

in France. LEMPmiCAL EVIDENCE.

Fire is the most fi'equent cause of business

interruption. If a damaged firm is iosured against

business interruptions, the insurance company

pays for the losses in sales, minus the costs spared because of the interruption of the production. Two facts motivate the analysis of

business ioterruption.

231
First, on a microeconomic level : tfie value

of a business interruption claim is seldom a

simple proportion of the size of the physical

damges due to fire. Sometimes its value is insignificant compared to the fire-damages, and

it may be much greater than the value of the equipment, machines, furniture or buildings

burnt. Second, on a macroeconomic level : the most striking feature of business interruption

claims is the extreme variability of yearly claims

recorded in a country. In France, for instance, the

total amount of business interruption claims paid

in 1988 by the insurance industry was nearly US

$ 200 million, that was twice as much as 1987's (US $ 87 million). And for the first time in the

history of that type of insurance in France, the

total amount of the claims paid to the firms was

greater than the premiums collected (US $193

million). The variability can be easily explained

by the occurence of huge claims, the amount of

which is greater than US $ 16 million and even

sometimes greater than US $ 160 million. They

are not numerous, but they make the tail of the yearly size-distribution of claims extremely long.

The analysis of that tail is the main subject is the main subject of the paper.

All the data used in our analysis originate

fi-om the statistics of the French insurance union : "ASSEMBLEE PLENIERE des SOCIETES

D'ASSURANCES de DOMMAGES" (APSAD).

it records all business interruption claims due to

fire, the amount of which is greater than US $

1600. All the claims are located in France, the

available data ^an the years 1975 up to 1991.

Yearly claims are ranked and valued in constant US $ (reference year 1988).

Fifteen out of the seventeen yearly size- distiibution (all but 1979 and 1981) look like tiie typical distributions shown on the graphs # 1 and

#2. The abscissa is the Log of the sizes of the

claims. The ordinate is the Log of the number of

claims which are greater tfian the size on the

abscissa. It is thus the Log of the complement of

the cimiulative distribution fiinction of the random variable ; "size of yearly claims". The

common slope of the parallel dotted lines is -1. The small steps on the curves are by-products of

the tendency to report rounded values of the damages.

N.B. Claims amounting to less than US $

16000 (=100000 F.) are not shown on the graphs. That threshold may be approximatively

considered as the median of the statistical distribution of claims : in 1988, for instance, 227

claims amounted to less than US $16000 and

223 amounted to more than this value. But the cumulated value ofthose small claims, below the median, is always insignificant. For instance, in 1988, 1989 and in 1990, their cumulated value

amounted to a mere 0.3 percent of the total value

of the yearly claims (in 1991 that proportion fell to 0.2 percent). Thus the loss of information is harmless.

Losses in 1983

\ \

N

\ :

10^ 10* 10' 10^ 10' Amount of Loss (FFr.)

Graph #1

Losses in 1986 ^^^^^^ \

\ A, \ X > jO \ \ \

o \ \ : \ \\

>

1o

t

, ' , , 10^ lO' lO' 10* Amount of Loss (FFr.)

Graph #2

232
Except for the years 1979 and 1981

(fortunately, no very large claims occured these two years), the size distributions of yearly claims

always display the same two features. Below a

threshold of about US$ 330000 the curves are concave.But above tliat threshold, the tails fit a

straight line the slope of which is close to -1. The

behavior of the tail can be described and the value of its slope can be approximatively

estimated by means of a graphical analysis.

Graph #3 shows the superposition of three size- distributions of yearly claims corresponding to years 1975, 1976 and 1977. The common

behavior is striking (in the next chapter we shall use a more rigorous estimation technique). The

straight line is also known in economic literature as the Pareto line. Each year the cumulated value

of the claims on the Pareto line amount to at least 80% of the total value of the claims. In 1988 and also m 1989 they even shared 92% of

that total value. Thus it is easy to understand why

the Pareto line is the heart of the matter for the iasurance companies. Their managers speak

about "lucky" years when no large claims occur,

as in 1979 and in 1981. Nevertheless those

happy years are exceptional, they give only a temporary relief Most of the time the claims on

the Pareto line strikes a severe blow to the profits

of the insurance industry.

1——1— 1 1 —

\ ; :

: \

"

>....xrL \

-

'1 1 —

10* 10* 10* 10* 10* Amount of Loss

Graph #3

N.B. It is worth noting that the size- distribution of business interruption claims in Westem Germany in 1989 also fits a Pareto line with a slope close to -1 . But the data recorded by

the COMTTE EUROPEEN DES

ASSURANCES in that country do not state the claims less than US$ 3,3 million. Hence the number of claims reported is small in absolute

value (32 instead of 1 12 claims larger than US $

330000 in France that year). Nevertheless the number of huge claims is relatively large in Western Germany, because the number of firms

insured in that country is greater than in France (where there were only 12 claims greater than

US $3,3 million the same year).

ILTHEORETICAL PROBABILITY

MODELS.

Since the data in our sample only record the yearly values of the claims greater than a

(relatively) high threshold, and since the size- distribution is LDvariant (except for two "lucky"

years), the theoretical model is to be found in the families of "max-stable" extreme distiibutions. One of the main theorem of the mathematical

theory of extreme values, states that there are only three types of max-stable probability

distribution fimctions,Ref [2], Ref.[3].

- The Pareto d.f : 1- x'^^ x > 1 a > 0

This d.f has a "heavy" or "fat" upper tail. Its main mathematical feature is that it has no

variance when a < 2 and no expectation when

a <1. The exponent a also gives the value the slope of the Pareto line on a double-log graph.

- The WeibuU or "type 11" d.f : 1- (-^T^

1< X < 0 a < 0 ; This d.f has a short upper tail. The Weibull d.f is found in the context of a maximum value that cannot be passed. This is not the case in the data concerning business

interruption : the maximum potential value of a claim is far greater than the record value already

observed (between US $ 2,1 billion and US $

18,7bmion!).

- The exponential d.f : 1- e"* x>0

This d.f. has a medium upper tail. In

practice, when dealing only with the large values

exceeding high thresholds (US $ 3,3 million in our samples), this outstanding result means that only one of those three d.f can be observed.

However, since both the cumulated distribution function of the Weibull law and of the exponential law display a concave tail without a

233
straight line when drawn on a double-log graph,

only the Pareto law may be relevant to the data on the business interruption claims. (With the exception of the two "lucky" years, 1979 and

1981, without major claims, that cannot fit the Pareto law, but may fit the exponential

distribution or the Weibull law).

m. ESTIMATION OF THE SLOPES OF

THE PARETO LINES.

Since each year we know the value of the empirical threshold (US$330000), the estimation

of each yearly Pareto distribution is completely

performed through the estimation of its characteristic exponent a.

Let N be the number of claims in the Pareto tail, X the size of a claim, xq the threshold value, i the rank of a claim with the value xj (i-l,2,.., N); the maximum likelihood estimator of a (or

Hill estimator, Ref|4]) is:

1/a = (1/N )2Log xj -LogXQ

We have E(l/a)= 1/a and Var( 1/a )= 1/Na^

Thus, the confidence interval defined by means

of the central-limit theorem is, for N sufficiently

large:

VNa(l/S -I/a)~D^ (0,1)

This means that there is a probability 0.95 that the tme value of 1/a lies within the interval;

1/a (1± 2/Vn)

Table #1 sums up all the estimations,

except for the two "lucky" years 1979 and 1981

when no huge claims occurred, thus changing the Pareto line into a concave curve.The last column on the right shows the values of the largest claim

(maxclaim) in thousands of US $, each year .

TABLE #1

ESTIMATED CHARACTERISTIC

EXPONENTS

Year N a conf.int. maxcl.

1975 29 0.9523 0.69-1.51 6789

1976 46 0.9283 0.72-1.32 18844

1977 40 0.8937 0.68-1.31 6611

1978 46 1.0250 0.79-1.45 7434

1979 29 _ _ 4685

1980 59 1.0130 0.80-1.37 11689

1981 48 _ _ 4315

1982 71 1.1117 0.90-1.46 18667

1983 68 0.9775 0.79-1.29 24116

1984 52 0.9918 0.78-1.37 27191

1985 61 1.0549 0.84-1.42 6584

1986 71 1.0072 0.81-1.32 16776

1987 61 1.0152 0.81-1.36 5549

1988 91 0.8854 0.73-1.12 24547

1989 112 1.0235 0.86-1.26 13183

1990 114 0.9049 0.76-1.11 52542

1991 127 0.9779 0.83-1.19 16076

Average value of a =0.9842

Median value of a =0.9918

All these estimations are compatible with

the theoretical value inferred fi'om the graphical

analysis of the data one. The sample

fluctuations are small (range of the estimated

values : 0.8854-1.1117) and there is no trend in the estimated values of the characteristic exponents. But, a trend obviously appears in the number of yearly claims greater than US $3,3

million. It is due to tJie increasing number of

firms insured. Nevertheless, no significant

correlation is measured between the number of

claims in the tail and the estimated characteristic exponents (R=-0.05).

A property of the Pareto law is its stability

when the extreme values only are recorded,

Ref [5]. This means that the size-distribution of

the n records of n identical and independent

Pareto laws with a minimum value m and a

characteristic exponent a=l is the same Pareto law with a new minimum value n.m. Here the records are the 15 "maxclaims", hence the

theoretical minimum value is US $ 5 million. We

can verify that the Hill estimator gives the same

characteristic exponent : a=0.9847. With other

minimum values, greater than US $ 5 nullion, we

have

:

234
a=1.0001 with m= US $ 5078000.

a=1.0006 withm= US S 6789000.

(but there are only 1 1 records larger than

this very high value).

N.B. In Western Germany in 1989, a= 1.0 149

with the confidence interval 0.75-1.57. The

largest claim amounted to US $ 240 million.

This huge claim is not far from the greatest

historical business interruption claim in France : US S 330 million. It happened before 1975. It nearly equalled two years of business

interruption premia paid in France. The second

largest historical busiaess intermption claim

happened in 1992, in an oil refinery. It amounted

toUSS ISOmilhon.

CONCLUSION.

Large business interruption claims in France

display a remarkable feature, their yearly size- distributions fit accurately a Pareto distribution with a constant characteristic exponent a around

one. This means that without an objective

ceihng, the theoretical probability distribution of

the claims have no expectatioiL Claims can have

a giant size, may be a size larger than the worst

claims akead)' experienced in the past. Those

huge claims show that business interruption is of

the same nature as natural hazards, for instance

hurricanes or earthquakes, it can be an economic

catastrophy.

REFERENCES.

[1] Zajdenweber D., Equite et Jeu de Saint- Petersbourg, Revue Econ.,l(1994)(in press).

[2] Galambos J., The Asymptotic Theory of

Extreme Order Statistics, 2nd.Ed. Krieger,

MalabarCna), 1987.

[3] Gumbel E.J., Statistics of Extremes,

Columbia University Press, New-York, 1958.

[4] Hill B.M., A Simple General Approach to Inference About the Tail of a Distribution,

Ami.Stat.,3 (1975),1 163-1 174.

[5] Mandelbrot B.B., New Methods in Statistical Economics, J. of Pol. Econ., 71(1963), 421-440.

Acknowledgement The author wishes to thank

MM. Sylvain Tribouillois and Fabrice Genest for providing the data on business interruption

claims in France, Mico Loretan for the graphs on

the size-distribution of claims. Financial support

from the "Federation Frangaise des Societes

dAssurances" (FFSA) is grateftilly

acknowledged.

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4 - 3 0 0-5 6 8 / 0 305 0 235

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