NATIONAL BUREAU OF STANDARDS REPORT

2190

A NEW METHOD OP ANALYZING EXTREME-VALUE DATA

by

Julius Lieblein

<NBS>

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NATIONAL BUREAU OF STANDARDS REPORT

KBS PROJECT NBS REPORT

1103-21-5106 13 January 1953 2190

A NEW METHOD OF ANALYZING EXTREME-VALUE DATA

by

Julius Lieblein

Statistical Engineering Laboratory

Research supported by the

National Advisory Committee

for Aeronautics

NACA Project W-5367

The publication, re

unless permission li

25, D, C, Such per

cally prepared If th

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

• In part, Is prohibited

Standards, Washington

•eport has been spec'fi

eport for Its own use
t

-
FOREWORD

This report develops various aspects of a

new method of treating extreme-value data® This method,

based on order statistics, has a number of advantages

over existing procedures, and will be useful in the

efficient handling of small or large sets of extreme

gust°load data and extreme data in many other fields

as well e

Ihe present report completes work under a

research project aimed at the improved application of

the theory of extreme values to the analysis of gust

loads of airplanes* This project, supported by the

National Advisory Committee for Aeronautics, was carried

out by Julius Lieblein under the general supervision

of Dr„ Churchill Eisenhart, Chief of the Statistical

Engineering Laboratory,, The Statistical Engineering

Laboratory is Section 11*3 of the National Applied

Mathematics Laboratories (Division 11, National Bureau of

Standards), and is concerned with the development and

application of modern statistical methods in the physical

sciences and engineering*

J, Ho Curtiss

Chief, National Applied

Mathematics Laboratories

A. Vo As tin

Director

National Bureau of Standards
.

/

.
TABLE OF CONTENTS

PAGE

1. SUMMARY 1

2* INTRODUCTION .............. 4

3. PRINCIPAL SYMBOLS ............ 6

4. STATISTICAL THEORY ........... 11

4.1 Extreme=value distribution and ' meaning of parameters ..... 11

4«2 Determination of method of

estimation .......... 16

4. 3 Order-” statistics approach for

small samples ......... 22

4*4 Extension to larger samples ... 33

5. SUMMARY OF PROCEDURES ......... 42

6 e COMPARISON WITH METHOD IN PRESENT USE . 58

6.1 Theoretical comparisoh ..... 58

6.2 UomparisdH based on a sample' ' ' oY actual observatiohs .... 63

6.3 Advantages and limitations of

proposed method ........ 65

7. CONCLUDING REMARKS ........... 68

REFERENCES 72
-

'
TABLE OP CONTENTS 9 continued

iv

PACE

APPENDIX A* Proof that sufficient statistics do not exist for the parameters

of the extreme-value distribution ® „ . , 75

APPENDIX Bo Mathematical formulation and

solution of, minimum-variance problem * * 79

APPENDIX C. Short-cut method for verylarge samples . . . • • * .

* . * • **•*•,*•• 86

APPENDIX D® Analysis of confidence intervals

in order-statistics method and method of

moments of Gurribel 93

APPENDIX Eo Details of theoretical comparison between order-statistics estimator'

and moment estimator of Gumbel » . ® . • 107
. . r.

' ,

‘
LIST OF TABLES

PAGE

Estimation of extremes

Worksheet 1 120

Worksheet 2 121

TABLE la Weights for minimum-variance „ unbiased* linear order-statistics estimator

L = E p 122

TABLE II o Means, variances and: covari^-ces

of order-statistics y^ 123

TABLE III. Variances and efficiencies of

minimum-vari ance * unbiased* linear

order-statistics estimator* L = Part A

r 124

Part B 125

TABLE IV. Efficiencies of order-statistics

estimators for various sample sizes

n = km+m 9 126

TABLE Vo Biases* mean square errors, and

relative efficiencies of order-statistics

estimator |p to Gumbel estimator 127

TABLE VI o Bias and efficiency characteristics of original Gumbel estimator,

la,n> and simplified (asymptotic) foting^ 128

TABLE VTI „ Probabilities of extremes

Part A. 129

Part B 130

TABLE VI lie Comparison of confidence

interval half-widths for extreme predictions given by Gumbel method, by

modified method, and by order-statistics

method 131

TABLE IXo Empirical sampling values of

first and second moments of sample mean

y and standard deviation, s 132
‘

,

.
LIST OF FIGURES

PAGE

FIGURE L General form of extreme-value

distribution (density function, f(x)) 133

FIGURE 2 0 Density function f (x) for ex- t treme-value distribution with parameters

p = l, u = o 134

FIGURE 3° Comparison of efficiencies of

order-statistics estimator | for

sample sizes 2,3*4 >5*6 135

FIGURE 4® Graphical analysis of a sample

of 23 maximum acceleration increments

by method of order statistics 136

FIGURE 5® Comparison of order-statistics

and Gumbel methods of analyzing a

sample of 23 maximum acceleration increments 137

FIGURE Go Comparison of empirical sampling

values of relative efficiencies of

order-statistics estimator to Gumbel

estimator 138
.

.
A NEW METHOD OF ANALYZING EXTREME-VALUE DATA*

by

. Julius Lieblein

1 o SUMMARY

A new method is presented and proposed for

analyzing extreme -value data which may arise in a

wide variety of applications <> Classical applications of statistical methods,

which usually concern average values, are inadequate

when the quantity of interest is the largest (or

smallest) in a set of magnitudes,, This is the sit<=

uation in a number of fields, e 0 g„, gust loads of

an airplane in flight, highest temperatures or lowest pressures in meteorology, floods and droughts

in hydrology* creaking strengths in materials testing, breakdown voltage of capacitors, and hpman life

spans, in all of which applications of methbds for

dealing with extremes have already been made 0 Discussion of the proposed method is preceded

by the necessary statistical theory (Section 4) which

also furnishes a basis for evaluating the new method

in relation to existing ones„ The techniques described provide a simple means for estimating the

*• Preparation of“this~report was sponsored by the

National Advisory Committee for Aenonautics 0
.
necessary parameters* making predictions from the

fitted curvej estimating the reliabili ty* and eval°

uating the efficiency of the method in relation to

other methodSo Moreover* these quantities are all

produced by a single set of computations involving

just two worksheets 0 This background material is

not essential to an application of the method* and

may be omitted if desiredo The method itself is

summarised for practical convenience and illustrated

s tep-by-s tep in Section 5 5 and compared with present

procedures in Section 6 » The latter section also

1 discusses the advantages of the proposed method*

chief among which ares

(1) For the first time there is available

an unbiased estimator of known

effi ciencyl

(2) The proposed estimator appears to be

more efficient than a simplified form

of the Gumbel estimator in many

practical cases* namely* for samples

The te chni c al terms used here are defined and discussed in the main text* and they can be located

with the aid of the list of symbols in Section 3®
.
3

of about 20 or move s and P = „95>

or more. The improvement in efficiency increases with increasing P or

increasing sample size„ When compared with the original Gumbel

estimator,, the proposed one is up

to twice as efficient,,

(3) The confidence intervals have a more

valid theoretical basis and are

in many cases narrower than the

ones in the Gumbel method,,

Included in the report are several appendices

giving mathematical developments not given in

the text
1

.

'
2 0 INTRODUCTION

The statistical theory of extreme values has

been found to have wide applicability in many diverse

fields, for example, meteorological extremes, floods,

drought’s, breaking strength of textiles and other

types of material?, span of human life, gust loads

experienced by an airplane in flight, breakdown

voltage of capaci torso

I The two existing methods of analyzing extremevalue data have several limitations, discussed in

the body of this reportc One of these methods is

known as the method of maximum likelihood and has

been described by B c F« Kimball (references 9? 10) 0

The other, the method of molnents, has been developed

by Eo Jo Gumbel (references 2, 3* 6), and its application

to gust-load problems has been discussed. in detail

in a previous NAGA Technical Note by Harry Press

(reference 18) 0
... .

.

. . .
5

The present report gives a new method for

dealing with the problem of analyzing extreme

measurements, treated ih the above Technical Note,

which has certain advantages over the existing

methods o The method of application is presented

in detail, together with the necessary worksheets

and other data, and the new method is compared with

the method of moments previously in use. For

definiteness, the discussion is at time presented

in terms of application to gust loads, but the

methods are also applicable to other fields where

extreme values occur c

3» PRINCIPAL SYMBOLS (Listed Alphabetically)

Note? By ^samples** are meant independent

random samples from the extreme**

value dis tribution

Equation no .

,

etp 0

9 9

X “ l^pcoo^n

Numerical quantities

entering Into weights

of order-statistics

estimator for sample

of n

(4.17)1

Table I

cov(y p s) or

o(y„s)

Covariance of mean

and standard deviation in samples of

n from reduced distribution

Page 110

(Appendix E)

E ( * ® e ) Mathematical expectation (or mean value)

Of (

* *;« )

Various*,

e.g., (4.6)

E „ E

m* n Efficiency of orderstatistics estimator

for subgroups' of m

observations , or for

samples of n

( 4 . 24 ) , (4-19)

Table m , Part B

E (s ) Mean Value of stand-'

ard deviation in

samples of n from

reduced distribution

Page 109

(Appendix E)

F(x) = F(xfu*>(3 )-

expt-e-

(x ' u)/P

]

Probability (emulative) distribution

function (c<,d.f ») of

extreme-value distribution with two parameters

(4.D

f( X ) = ^

Density (or frequency)

function of extremevalue distribution

F(x)

Page 11;

Figure 1
•

-

.
7

k Number of equal sub- Pages 33 * 36

groups of size m

contained in sample

of n.

m Size of one of the k

equal subgroups contained in sample of n

Pages 33, 36

m 8 Size of remainder subgroup in sample of n

that is left after k

equal subgroups of m

are taken 9 i 0 e 0 s n = km + m8

Page 36

MSE ( . . .

)

Mean square error of

(«..)> equals variance

plus square of bias

(4-13)

n Sample size (N denotes sample size in

Gumbel method)

Page 23

P Probability level

associated with a

predicted value

Page 13

Qo Numerator in CramerRao lower bound!

= Q° /n

(4-24)

Table III,

Part A.

^LB Cramer-Rao lower

bound to variance of

unbiased estimator of

parameter (see

Qo>

Page 26

%i 9 Qn

Variance of orderstatistics H sub»

estimator” for subgroups of.m, or of

estimator for sample

of n„

(4*23><r

(4.i8;

8

r Rank of r-th observ

vation (counted from

smallest) in samples

of n whqn arranged in

ascending order from

Smallest to largest

observation

h(t1s t

2

) = MSE(T2

) MSE ( T ) n

'/

1 2 (y.

n i=l ] y)‘

fjtelatiye efficiency of

es timator-x to T

2

(greater than unity

when T-, is mor^

efficient)

Standard deviation

(sodo) of sample of

n from reduced distribution

2 (xo =x)^

i=l

S 0 d 9 of sample of n

original distribution

^, 1 =1,2 ^

0 O O £ k

k

2 To

i=l

Order statistics

w sub-estimator” for

i= th of the k equalsize subgroups in

samples of n

Average of the subestimators for the k

equal-size subgroups

T° Sub-estimator for

remainder subgroup

(see m®

)

t,t» Weights for T and T®

in grand estimator

£or samples

= tT + t

s T®

original

Gumbel ® s/es timator of

mode u for sample of n

Page 54

(E.7)

Page 109

(Appendix E)

(4.11)

(4.21)

(4o22)

( 4 . 25 )

( 4 . 26

)

» (It. 27)

(E.l)

c

>

9

! = Xx -V^ "^^x Simplified expression

used to represent

Gumbel 8 s estimator of

mode u

(E.2)

u Mode or location

parameter of extremevalue distribution

Random variable

(^unreduced”

)

having extreme-valiie

distribution F(x)

Page 12

( 4 a)

( 4 . 5 )

x

l* x2* ° ° ° s ^n

Xi , X , X , j An pr Vn

0< /( <ju< V <1

7s Jj

P

The n order-statistics Page 23

in sample of n, i,e # , the observations ranked

in ascending order

Three selected order- Page 86

statistics in Mosteller (Appendix C)

method for very large

samples of n

Sample mean in sample Page 20

from original ( Mun=

reduced’®) distribution

Reduced variate Page 11+ Scale parameter of Page 12

extreme -value distribution F(x)

A s

p

8 = s K H X

{ = 0o57721,

56649

original

Gumbel 8 s/es timator of (Eol)

(3

1 for sample of n

Simplified expression (E 0 ?)

used to represent

Gumbel 8 s estimator of

P

Euler 8 s constant (I4.08)
,

.

*
10

A

x,n

= lo1

^

A

9 = 1.11+lBpP

fi = E(x)

fi2 - o h

A

io

o

2

(s)

y

s u + py.

ax

'

f(x) = “J

J(y) = exp(~e”^)

Half-width of 68%

confidence interval

in Gumbel method

(D.l)

Half“Width of 68 % (Doll)

confidence interval

when mqdified by

probability factor

Half “Width of 68 Jo

confidence interval

in method of order

statistics'

First moment or

mathematical expectation of random variable x

Variance of reduced

dis tribution

The 100P-percent

point of the extremevalue distribution F(x)

Simplified expression

used to represent

Gumbel estimator of

ip

Variance of standard

deviation in samples

of ^n from reduced distribution

Population variance of (4»7)

x

1 * y

Plotting position of Page 5>4

r-th observation ranked

from smallest

Table VIII

(4.6)

(4*9)

Page ll|

(E.3)

Table IX

Cod 0 f» of reduced

extreme-value distribution

Page 112

(Appendix E)

4<» STATISTICAL THEORY

4.1 Ex treme-value distribution and

meaning of parameters 0

The method of analysis presented herein is

based upon the assumption that the observed maxima

to be analyzed are independent observations from a

statistical distribution pf the form

F(x) = F(xsu,0t) = exp(=e"

(x “ u)/!3

) c ( 4 . 1 )

This is the cumulative (or ogive) form of the distribution* which expresses the chance that an observed

extreme value (gust load* for example) will not

exceed x in value. The more familiar concept of

frequency or density function* f(x) - F'(x)* for

this distribution may be obtained by differentiation

but is rather cumbersome (see Appendix A) and is not

needed for present purposes® The general shape of

the density function f (x) is shown in Figure 1 0 The

meaning of the various quantities indicated is explained below, A more detailed graph for the case

where the parameters are u = 0* [3 =1 (the ‘‘reduced'*)

extreme-value distribution) is plotted in Figure 2 0

The distribution (4»1) has been studied extensively by Gumbel (among others) (references 2*3*6) and
'

.
12

is known as the asymptotic distribution of largest

values o We shall refer to it briefly as the ex tremevalue distribution ,, The significance of the term

asymptotic" is as follows ft* If the underlying distribution of all (not merely the largest) gust loads

(e 0 goj, effective gust velocity, normal acceleration)

is considered, then the largest values in repeated

large samples from this distribution have a distribution of their own which, as the sample size becomes

larger; and larger, approaches closer and closer (in

a certain sense) to a limiting distribution,. This

limiting distribution is, according, to evidence

$T3 - • presented in reference 18 , of the form (I4

. 0 I), with

1/(3 replacing the parameter a used in the reference

„

The parameters of the extreme -value distribution are depicted in Figure 1„ The quantity u is

the mode or highest point of the (frequency) distribution,, The quantity (3 is a scale parameter,

analogous to the standard deviation a in the case

of the normal dis tribution,, In fact, (3 equals

(about 3 /i|) times the standard deviation of the

extreme-value distribution,.

Although the two parameters u, (3 completely

specify the distribution, it is desirable

13

to introduce another quantity j* =u + py which is a

linear combination of the parameters u and p

(and therefore* since we shall assign known values

to y* itself a parameter) * and makes it possible to

estimate u and p simul tapeously* rather than in terms

of two separate problems 0 Thus if we can estimate

as a + by with a* b known * then we can read off at

once the values u = a* p = b Q

The parameter^ has another highly important

meaning o In Figure 1 the area P under the distribution to the left of the ordinate erected at J represents the probability that a value larger than ^ will

not occur e If J

is very large* then P very nearly

equals the whole area? unity* which means an observation is almost certain not to exceed fg

j in other

words* a larger value of i will occur only very rarely,,

Thus if P = o99i> then the corresponding value of

has a chance of only O 01 of being exceeded 0 To

denote this dependence of/ upon the probability

-:$ That is* we are concerned with the transformed

parameters ( ^ *p ' ) * obtained from the original

parameters (u*(3) by the linear transformation

|~u4-py *p

8 =B„ We shall henceforth give attention

only to the first parameter J * disregarding the

second parameter p

c of the transformed pair (i*p 8 )c

Whenever it should become necessary to refer to p

? * however, the prime will be dropped for simplicity,,

(See second footnote to page 26 )

14

P we use a subscripts C c This parameter is called

^ P

a percentage point or the lOOP-percent point of the

(extreme-value di stribution) 0 If we can estimate l

^P

for different probability levels such P = o90 s ®95?

e 99> etc 0 * then these values are precisely the predictions we desire for* say* gust-load accelerations

that will be exceeded (on the average) only 10*

1* etCoj respectively^ times in 100 o

The explicit relationship between^

p

and P

can be determined by means of formula (U°l) for the

extreme-value dis tribution „ If x is put equal to

Jpj,

then P* the probability of not exceeding this

value* is simply F(|

p

) 0 Thus

= (£ -u)/S ~

P = F($

p

) = exp (-e P

) = exp ( -e

Y

)* (4„2)

since ^p = u + py* Hence* for a given (usually

large) probability P* the corresponding J> p

is obtained

by finding y from the relation (4 o2)„ and then writing

i P = u + p ( 4 - 3 )

where the subscript P has been added to y to denote

dependence on.P, Comparison o,f the right members

of (I4

. 0 I) and ( 4 0 2 ) shows that the quantity y bears

the following simple relation to the corresponding

variable x in ( iq 0 1 ) s

or

x s u + (3y 0 (h°5)

Also* if in (4.1) we set u = 0, (3 ~ l s then x

has the same distribution as given by the right=

hand side of (4 „2) 0 In other words 9 y as defined

by (4.4) or (4.5) has an extreme-value distribution

whose parameters have the extremely simple values

u = 0 ? (3

1 = lo Thus y is called the reduced variate ”

and is pferfectly analogous to the standardized

variate t = (x- yu)/a of normal distribution theory.

The distribution of y in (U<>2) 9 called the reduced

distributions has been tabulated in Table 2 of

reference 1% 9 which also contains a table of the

inverse function as well as a number of other tables

The variate x is sometimes referred to as the

original or wunreducedw variate.
, • :•

'' v- • . d. . '
16

related to application of ex treme-value theory,,

From the above discussion it is evident that

the solutions of both the problems of estimation

and prediction are embodied in the one quantity

J,p = u + Py^o Estimation of this quantity will

be the main objective of the remainder of this report,

4 0 2 Determination of method of estimation

To avoid confusion, we distinguish between

a function of sample variables x^ ,

? « « -> » x

n

*

such as the sample mean g (x^0 ° * x

n

) s x

= (x,+x„+.,o.+x )/n= and the numerical values

1 2 n

' 9 o o o gQ

85 g(XpX2

S .o.‘1) x

n

) assumed by the function when

o the actual values of the observations x^ = x^ are

substituted into the function. If the func ti on is

used 'to estimate a parameter, we shall call it an

estimator of the parameter? the particular nume ri c al

value assumed in a given case shall be called an

estimate * In searching for estimators the first step

is to seek what are known as sufficient statistics , A definition of this concept may be found in any
•: ; - .

'

.
advanced text on statistical theory, for example

reference 8,Vol 0 II, p„ Qlj but the feature of importance here is that gjiven a set of joint sufficient

statistics, that is, certain functions of the sample

observations, it is often possible to deduce frolh thefa

an estimator with certain desirable properties, provided that the number of such functions does not

depend upon sample size,, If it turns out that the

only set of sufficient statistics is the trivial set

consisting of the n functions t^ (x^ , „ . » ,x^ ) = x^,

i=l, ooo, n, i 0 e 0 the n sample observations themselves, then obviously this furnishes no guide whatever for constructing fline ti 6ns of the x*s which are

optimum estimators.

Investigation reveals that, unfortunately,

jDirit sufficient statistics do not exist for the two

parameters of the extreme rvalue distribution, A

probf of this fact, (which was conjectured by Bo F„

Kimball, reference 9? P« 299) has been discovered

by lo Richard Savage of the Statistical Engineering

Laboratory of the National Bureau of Standards, and

is presented in Appendix Ao
.

-
18

It may be note.d that Kimball ( i bid „ ) has studied

i a broader concept called "set of statistical estimation

functions” whereby the estimators of the parameters

are given, not by explicit formulas involving only

the sample values, but implicitly as the solutions

of a set of simultaneous equations, for example,

the classical maximum-likelihood equations 0 Unfor -

tunately, such estimators do not seem to lend themselves to the procedure referred to above for constructing optimum estimators, and there seems to be

no analytical means of accurately evaluating the

important characteristics of bias and efficiency,

defined below, for such estimators In the case of

f ini te samples „ (Although these estimators may be

asymptotically optimum, i 0 e 0 , for infinitely large

samples, this need not be the case for samples of

finite size.)

A second method of approach to the problem

of estimation is the classical one known as the

method of moments 0 In the case of the extremevalue population this method is as follows 0 The first two moments of the extreme -value

population (4«1) are
1

. 0- jR|9

• s
19

f*x

~ E(x) = u + (3E(y) (4.6)

= E[x-£(x)] 2 = (3 2E[y-E(y) ] 2 = (3 2

o

2

(4.7) x y

where y has the reduced ex tr erne ~ value distribution

(4«2)p E denotes mathematical expectation* and a is the variance* the second moment about the mean e Using the moments of the reduced distribution: (see,

I *. e„go, reference 8* Vol 0 I* page 221)

E(y)

y

we obtain

s if - <>577216 (Euler's constant) (4»8)

^2

l c 644934 (4-9)

u

' + °x =JB P (4.10)

relations which express the population moments in

terms of the population parameters 0 Therefore* if

we had good estimators of the population moments*

we could readily find the parameters c This fact

constitutes the essence of the method of moments

„

It consists in treating the sample as an adequate

representation of the population* replacing the

population moments in the expressions which relate
fi
20

them to the parameters by the corresponding sample

moments , e„g 0£) by the sample mean x, and ox by the

sample standard deviation

sX

TT”

2

i-1

(x, -x)

( 4 * 11 )

This gives x = u + ^fp, s = (n/y^Ps which

X

yield the moment estimators of the parameters;

for p # p = j/&A)sx j

for u, u = x ^A)sx

* (4ol2)

These are essentially the estimators which form

the basis of Gumbel s s method (reference 6» Lecture

A A

3 s> eq 0 (3o29) p with - u, l/a “ P)

w

* Thi s method

is justified by the fact that under general conditions

a a , the estimator functions u - u(x 9 x 9 „ „ 12 0 s x ) , n

A A

P = p (x^^x^ii * o »

x

n

) in (4»12) approach(in a certain

sense) the values of the corresponding parameters

u 5 p as the sample size becomes infinite 0

The actual estimators used in the Gumbel method

are slightly more complicated (reference 6 g Lecture 3* eq, (3<>39) s> but the difference is

not important at this pointo (See Appendix E)»
'
21

This method has apparently given satisfactory

results in practice, It is, however, subject to an

important limitation. In studying estimators it is

highly desirable to know something about their probability distributions--!! not the exact density functions, then at least their means and variances. The

mean value (mathematical, expectation) of an estimator

indicates whether on the average the estimates given

by it are too high or too low relative to the actual

values of the parameter estimated --in other words,

whether there is any bias in using the estimator.

Similarly, the variance indicates how much the

estimates scatter among themselves and is the basis

for constructing a measure of efficiency which makes

it possible to compare the performances of different

estimators, A more useful concept for some purposes

than variance is mean square error (abbreviated MSE)

which measures how far the estimates deviate, on the

average, not from their own mean, but from the quantity

the parame ter°=whi ch they are supposed to measure.

There is a simple relationship between variance and

MSE, namely,

2

Mean square error = variance + (bias) ( 4 . 13 )

22

Thus* for unbiased estimators * variance and M&E are

identical * and for brevity we shall use thb term

r ’’variance”' in such cases. But it should be remembered

that the concept in view is actually the MSE, This

becomes especially important later when biased estimators are d iscpssed (Appendix E)* and variance and

MSE are no longer identic al „ If we try to determine the mean (or expected)

values of the estimators* u* p*, in (1|<>12) we find

that statistically these functions are quite complicated*

leading to very difficult multiple integrals which

apparently can be evaluated accurately only by 1 arge~

scale numerical integration. This difficulty evidently

persists if we are interested in the parameter

|p = u + (3yp

instead of u or p separately*

4 ,3 Order-statistics approach

for small samples

Apparently the qnly method of estimation which

avoids this difficulty is the method of order statistics ,

* Shorter methods of limited accuracy are possible

and have been used in this report for comparison

purposes, (See Section 6»1 below and Appendix E,)
.
23

If the values in a sample of n observations are

arranged in* say, increasing order of size, and deno ted by x , x 0 ,

. o o , x , x1 < x0 < o o o , < x .then

L CL “ n J these values x^ are called order statistics ,. The

smallest is called the first order statistic! the

middle one (if n is odd)*, the median! the one which

is one-fourth the way up from the bottom, the first

quartile, e tc 0 (If there are several equal ones then

suitable modifications are made in the defini tions )

„

There is an extensive literature on this subject*,

chief among which is the comprehensive survey in

reference 20 o Order statistics provide rapid and practical

methods of analyzing data® The range, x

n

° , is

a very common illustration from quality control <>

It is simply the difference of two order statistics,

the largest and smallest, and its properties have

been extensively studied for samples from the normal

dis tribution» The range has been found to yield

estimates of the standard deviation of the population

that often compare very favorably with the theoretically

best obtainable o More general linear functions,

* C^Xt, + G^X2

+ 000 + c nxns> give weight td every

214-

sample value 9 have also been studied (reference 19 ) s and values of the coefficients have been found which

make it possible to estimate very simply and remarkably well certain quantities which previously were

obtained only by more complicated c alculations

„

We shall carry over and extend this procedure

to the case of samples from the extreme -value distribution (4.1) . The method will in many respects follow

the general approach used in reference 19 for several

other distributions „ The aim is to determine the

weights i-1 ,

„ «

„

s n 5 for all the n order statistics

in a sample of size n so that the linear estimator

n

L = S w„x4 ( 4 ® 14 ) i=l 1 1

has the properties we desire 9 namely

(l) The mathematical expectation equals the

parameter to be estimated s i,e. the estlmator is unbiased g

1

E(L) - | (4.15)

P

The MSE s which in this case is the same

as the variance^ is as small as possible^,

consistent with ( 1 )

( 2 )

25

MSE(L) = o2

(L) = E[L=E(L)] 2

= a minimum# (4ol6)

An estimator L which satisfies these two

conditions will be denoted by^ps a notation suggested

by condition (1 ) „ Condition (2) is equivalent to saying that

the estimator £is as efficient as possible under

the given conditions 0 This concept will be discussed

below#

The mathematical formulation of this minimum^

variance problem is developed in Appendix B p and the

solutions (the weights) are shown in Table I for

n - 2 to n - 6# The case n greater' than 6 is dis=

cussed in the next Section,, For each given value

of n 9 n weights w

j_ 5 w2 s °°»s w

n

are determined that

depend on the quantity y^ that occurs in the parameter

£p - u + Pyp to be estimated# The w^ are each of

'

i the following forms

^i ” b .^yp

p i s lj 2 S

o o o p n o ( Lf. o 1 y

)

Substituting these weights for given n into (lf.ol6)

actually gives the minimum value s that the variance

can attain under the above condi tions* and this value

•depends upon quadra tic ally §

26

V , - Q

min n

= <A

nvB

n

y

p

+c

n

)p

2

( 4 . 18 )

Table I gives the values a^ 9 b^„ A^, B „ C , which

have been found by exact computation methods as

indicated in Appendix B and Table II there described,.

The quantities V . - Q are shown in Table III. min n

Part A.

As sample size increases , the estimation is

expected to improve and the variance to diminish,.

In order to have a convenient standard of comparit ft son, in the case of unbiased estimators, we scale

all variances by dividing into a theoretically specified

variance ( Q_ , known as the ”Cramer~Rao lower bound”

(reference 1, pp® 4^0? equation ( 32*3 <>3a) which

Tc For biased estimators, see Appendix E®

JHS’ This Cramer-Rao bound is given for the case where

the distribution has only one parameter to be

estimated® For the extreme -value distribution with

the two parameters (J, p) we can regard 6 as a

^nuisance parameter” and thus obtain a "Cramer-Rao

bound” for § , the expression for which will

involve (3 (see first footnote to Table III)® This

procedure is based on the ^method of nuisance

parameters” discussed in reference 11® (See also

footnote to page 13®)

2?

is less than or at most equal to the variance of

any (unbiased) estimator of the parameter in question"' o The result is then an absolute number

l .... - between 0 and 1 which, when expressed as a percentt age 5 is called the efficiency of the estimator for

samples of ns

Efficiency (L) = E

n

(L) s ® (4»19)

The quantities E , whiclp. evidently depend upon y-.,

n e and therefore upon P, are given for n “ 2 to 6 for

selected values of the probability P in Table III,

Part Be Part A contains the numerical values of

I the variances

2 of the parameter (3 o The expression for has

been implicitly given in reference 10, p„ 113, and

is indicated in the first footnote to Table III# Part A

of this repprto

and the lower bound in terms

There may or may not exist estimators whose

variances reach the lower limit Qt R

o If (as

may happen) there exists a Q,

9 > Q.H; such that the

variance of every estimator is > tp (and of course

» Q_ )„ then Q

9 may be substituted for in

the numerator of the above expression for efficiency

(4ol9) without the fraction exceeding 1„ The

investigation of the existence of Q,

9 is too complex a matter for purposes of this reports However s the only effect of using a lower bound

which is too low is to understate the efficiency,

so that the results are on the safe, conservative

side*,

28

Two points should be noted about the choice

of probability levels shown in Table III 0 The value

P = *,36788 ~ l/e, which corresponds to = 0, is

important because it gives the mode, one of the

desired parameters of the distribution*, This is

evident from the fact that the parameter we are

estimating is = u + py^

” u, the mode* for y.p - 0.

Similarly, the limiting value P = 1 corresponds to

the scale parameter (3*, This may be seen as follows.

If P approaches 1, the £ ^ and y^ both become indefinitely large, but their ratio £ - g /y = (u/y )+p

, P P P P

may be considered to be a new parameter which approaches

(3, since the mode u Remains fixed and finite (as does

also (3 ) o Hence we may estimate (3 by first estimating

gp

for arbitrary P and then letting P approach 1.

Now from (4»l4) and (4»17)j> the linear estimator

A

L = £p is of the form

i

p - f*i + yp

f

2 (4.20)

where f and f are functions of the sample values

which do not involve yp. By the preceding remark,

we can then estimate the parameter (3 by writing down
.
29

the corresponding estimator of gi ,

%' _ i_ " >? —

f I + f 0

yP

2

and letting P approach 1 9 obtaining

a

g

= t 0

P 2

as the corresponding estimator of (3#

A

In other words *, an estimator g of (3 may be

p

obtained by simply taking the coefficient of y

p

in

gp

when written in the form (4*20) 0 Similarly*, the

a 2 variance of is the coefficient of y

p

in the variance

of gp„ This may readily be seen as follows# From

(4*20)*, we have

°

2

(£j = o2

(f 1

) + 2y

p

cov (f s f^

)

1

2 2

+ y O

P 2

2

A + By

p

+ Cy

p

where A*, B*, C are quantities which do not involve

y

p

(though they may involve (3*, in general)^ thus*, as

P approaches 1 and y

p

increases without limit*,

? / A oil 1 2 *

2 ° y

p

A

+ C C
.
30

2 2 a the coefficient of y in a (g ) o From this it follows

/\ also that the efficiency of the estimator being

a ratio of variances, is simply the ratio of the

2 coefficients of y^, the other terms being disregarded.

A These facts applied to the estimator | make

P

it possible to avoid a separate treatment for the

two parameters u and (3 0 Their estimator are each

represented by a single line in a table (such as

Table III) showing values for various probability

levels :

P = o36788 (or yp = 0) gives u| P = 1 (or yp = 00 )

gives (3 »

The concepts of variance and efficiency have

also a more concrete, practical significance 0 The

lower bound to the variance has the form

Q_ ~ Q /n s where Q, is a quadratic function of y , ijB o o P

but is independent of sample size n„ For two samples

of sizes n'and n

n

, the variances Q

! « Q* are in

LB LB

Q

1 LB

LB

the ratio

31

i.e,, inversely proportional to sample size„ Similarly,

if we had two estimators for the same sample size,

we could form the ratio of their variances and think

of it as representing an (inverse) ratio of (hypothe ti cal) sample sizes » Thus, if for a sample of

20, the variance Q» of one estimator were one-half

the variance Q,”' of an alternative estimator, then

the first estimator would require a sample of only

10 to give as much information as could be obtained

with the second from a sample of 20 o This saving

of half the number of observations is expressed by

saying that the first estimator is twice as efficient

as the second o In general, a saving of the fraction

p of the observations makes one estimator l/(l-p)

times as efficient as a second,

A- The efficiencies of the estimators gpin Table

III are more conveniently compared in graphical form,

as in Figure 3, The heavy horizontal line at the

top indicates perfect or 100 percent efficiency, and

the rising curves as n increases show how closely the

estimator* is approaching the standard of perfection,.

The most outstanding fact is that, in marked contrast

*to a theoretical, perfect estimator, the efficiency of

32

A

the actual estimator: depends upon the probability

P, being largest for the middle ranges .i;0 to .60

and dropping considerably at the ends near 0 and 1,

Since analysis df extreme (largest) data is concerned

chiefly with the larger magnitudes associated with

very small probabilities of occurring or being exceeded

we shall limit our interest to the range above P = .90.

For n = 6 the efficiency exceeds the 80 percent level

for all values of P in this range that are apt t©

occur in paractice (i.e., P < o 999 )

In view of the

satisfactory values of efficiency* further calculation

for n > 6 did not appear warranted at this time,

particularly since it became apparent that the labor

of computation would increase out of all proportion

to the rapidly diminishing improvement in efficiency.

Of course, most samples of observations are

larger than the trivial size of 6* and the question

arises of how to handle the larger samples. This

is treated in the next subjection.
-
33

Extension to larger samples

The key to handling samples with more than

six observations is to treat them as sets or subgroups of samples of 6 (or if necessary, 5)° If a sample size is not an exact multiple of 6 or of 5?

then the sample may be treated as consisting

either of subgroups of 6 with an odd group remaining

having less than 6 items,* or of subgroups of 5 with

a remaining group of 6« We first deal with the simpler

case where n is an exact multiple of 5 or 6,

C ase I o Sample size an exact multiple of 5 or 6 , Suppose* in general* n = km, where m is the size of

subgroup, which need not be 6, and k is the number

of subgroups in the sample » If the sample Is so

divided into subgroups that the observations in one

subgroup may be considered to be statistically independent of those in any other subgroup, then it is

legitimate to treat the sample as consisting of k

independent subsamples, each of size m 0 One way of obtaining independent groups is

by use of random numbers 0 This, however, will lose

valuable information embodied in the order in which

’the data were actually observed 0 If the data are
'
truly random, so that, for example, there are no

seasonal effects, then this implies that subgroups

formed in the order in which the data are observed—

the first m values observed put into the first group,

the next m into the second, etc®— should be independent®

This assumption, of course, underlies the entire method

of estimation described in this report, and we shall

adopt it in our procedures®

From each subgroup we form the "sub-es timator ,s

m

T = 2 wx, , i = 1, 2, » 0 ®, k , (4.21

)

i j=l J J where the weights w-^, W

2

, ®®®, are those taken

1 from Table I for sample size m and are the same for

each subgroup of m values (but, of course, are different

for different sizes m) ® Thpse k sub ~ estimators T^

are then combined by simple averaging to form the

grand sample estimator?

1 k

T = = 2 T. , (4*22)

k i=i 1

The variance of this estimator is simply
'
35

since we are taking the variance of a mean of k

independent quantities T^, each of which has the same

variance'* Var(T^) = Q,^| denotes the variance tabulated in Table III, Part A, for m = 2, 3, 4* 5* 6°

The efficiency of T is, since n = km.

%,B E5 Qo 1

Qo

E = mmr

i = ”T“— = = E

Var (T) Qm

m

( 4 ° 24 )

Where Q^g = QQ/n - QQ /km, and Qq

is independent of n,

Thus we have the important fact that if a sample is

broken into equal-size subgroups, the efficiency of

the order-statistics estimator depends only upon the

size (m) of the subgroup (and, of course, P) » Since, according to Table III, Part B, efficiency

increases with sample (or subgroup) size, it follows

that when there is a choice, a sample should be broken

into subgroups as large as possible for best efficiency,

i 8 e, into subgroups of 6„ If this is not possible,

but if the sample size n is an exact multiple of 5*

then subgroups of 5 may be used with not much loss in

x These variances are equal because they depend only

upon m, P, and (3, which are constant for all the

subgroups of the same sample.
.

.
36

efficiency. The last two columns in Table III,

Part B, show that the loss is 2 0 i|. percent

(= o8647 ~ 08404 ) at P = c95j> rises to a maximum

of 3 08 percent for the limiting value P = 1

Case II «

1 Sample size not an exact multiple of $ or 6 „ In most cases, of course, the sample size will have

a remainder when divided by both £ ar>d 6 0 There are

then a great variety of choices as to how to partition

n into subgroups of 6 and 5> and perhaps other sizes.

Many 'of these possibilities have been examined, the

aim being to establish as simple rules as possible

without too great a, loss in efficiency. Fortunately,

most of the methods of partitioning a sample of given

size n do not lead to greatly different efficiencies.

Thus the following rules can be laid down for n > 7

1 (n <; 6 does not involve breaking into subgroups) based

on writing n in the form either 6k + m® or 5>P + 6,

where m* < Gt

(a) n = 7 up to large values ,

(!) Use n = 6k + m 8 s split up into k

subgroups of 6 and a remainder sub°

group of m 1 <6 items, unless n = 31 # 61,

etc,, i,e,, a multiple of 30 plus 1

37

(ii) If n = 3Qk + 1, write it as

n = ( 30k~ 5 ) + 6 = (6k-l) x 5

+ 6 , i.e, split sample into

(6k-l) subgroups of 5 ana a

’’remainder” subgroup of 6 0

(b) n extremely large 0 If sample size Is

j of the order of several hundred or more,

so that the number of subgroups is of the

order of £0 on 100 , then the amount of

computation becomes increasingly lab~

orious o For such very large samples of

extremes, which are rather rare, a short=>

cut method is available which is explained

in Appendix C e While its efficiency is

. i substantially less then the longer method

presented here, it is nevertheless of

practical value inasmuch as the loss

in efficiency, which in practical terms

means an effective loss in number of

observations, is not very important when

a very extensive amount of data happens

to be available

38

The variance and efficiency of an estimator

for most sample sizes. (Rule (a)) can be discussed

readily in general terms » Assume that n = km + m’

represents the separation of the sample into two

parts s one consisting of k equal subgroups of size

m = % or 6; the other consisting of the remainder

subgroup of size m f < m except for the exceptional

case where m = 5, m' =6 (Case II, rule (a) (ii) )

The average, T, is formed from the first part as

described under Case I, Then a sub=es timator T' is

formed from the remainder subgroup of m 1 values using

i the weights w^ for samples of size m'

m* i < T* = 2 w‘x. , . (4o2£)

i=l 1 1

j where x. , i = 1, 2, m ! , denotes the m* values

i * in the subgroup e Finally a weighted average of T

and T ! is formed, and this is the grand sample

A

estimator

f = tT + t'T» v ( 4 » 26 )

39

where the multipliers •St- are

( 4 * 27 )

Since all the subgroups are independent, sc are

T and T ? ? whence

Var (f

2

t> Q.m

1 since the variance of the mean T is » 0 9 k m

Prom the above it is evident that once the

partitioning of sample size n into n = km + m' is

determined, the variance and efficiency may be

obtained except (in the case of the variance) for a

factor (3^ which must be estimated from the data.

Table IV lists for convenience the efficiencies at

two probability levels, P - *99 and the limiting

value P = 1, for most of the sample sizes that

Sc Other multipliers are possible-. In particular,

there is an optimum set of multipliers which nroduces

an unbiased estimator fp with slightly smaller variance,

and hence slightly greater efficiency. The optimum

multipliers are, however, less simple than the pro=-

portions! ones- for example, they are not constants

but depend on P=— and the gain in efficiency is not

great® This was shown by a number of trials and by

the fact that In any event, the efficiency cannot,

exceed that for the larger subgroup size, E (or En , |fm ! >m), and does not differ much from It if the

total sample size n is at all sizable , say > 20®

may occur in practice t^i th gust-load data, provided

the sample is split up according to the above rules

The levels P = ®99 and P = 1 furnish a convenient

basis for comparing the efficiencies of two different partitions of the sample size® At this end of

the probability scale the difference between the two

efficiencies decreases monotonically as P decreases®

Thus,, if the difference in efficiencies is 3 percent at

P - ,99 and 4 percent at P = 1 , then the difference

is between 3 and 4 percent at P - ®995# say, and at

P = ®95> and under is apt to be substantially below

3 percent, a difference negligible for practical

purposes® The partitions shown in Table IV are those

recommended, by rule (a) above® In certain eases, the

efficiencies of alternative partitions are shown in

the footnotes to Table IV for use in case the extra

few percentage points in efficiency are considered to

be worth a little loss of simplicity in computation®

There are some useful a priori guides for

judging the efficiency in any given case .even beyond

the limit n = 4^ of Table IV® Thus, if n = km + m 5 , it is clear that the efficiency cannot exceed that

for the subgroup sizes m and m s , but must lie

kl

somewhere between the efficiencies corresponding to

these two sample sizes® If m and m

9 are not far

apart* then* regardless of the number of subgroups

k, the efficiency Is determined between narrow limits®

Again, if k is substantial, say near 10 or more, then

the efficiency is practically that for the larger

sstfnple size m® Of course the maximum efficiency

obtainable by the procedure outlined here is for

Case I when the sample size is an exact multiple of

6« For P = *99 the efficiency in such case is 83*2,

and for P = 1, it is 76*8* If any given partition

results In efficiencies within, say, 2 or 3 percent

of these values, then there Is nothing significant

to be gained by using any other partition, unless

it is such as to simplify the computation*
'
SUMMARY OF PROCEDURES

The above method of analysis will now be

summarized for ease of reference® The use of the

method has been considerably simplified by the construction of speci ally designed worksheets, represented

by the pair of blank specimen forms following this

page, A completely filled-out pair (Worksheets 1

and 2) will be found immediately preceding the tables

at the end of this report® With the aid of such worksheets about two hours should be sufficient for all

the calculations for a moderate-size sample, such

as the sample of 23 observations analyzed below, and

It has been found that this period is even sufficient

to include the graphical analysis also presented®

The materials needed for application of the

method, besides Worksheets 1 and 2 and a sheet of

extreme probability paper, are, in the order in which

needed 2

(i) Table IV, showing efficiencies for

various methods of splitting sample

into subgroups

„

(ii) Table I, giving the weights a^, bi,

(iii) Table III, furnishing the quantities

Q.023456 9 Q0 9 Q.- 1 Q, 9 Qj-.* Qx*
' ' ~ 1-

' 'Mn

....

•

' iV-iV.PV

it l
U3

ESTIMATION OF EXTREMES

Worksheet 1 - Determination of Estimators

Sources Computers

Dates

Record

No. Observed

extremes

I. SUBGROUP SIZES AND PROPORTIONALITY FACTORS

n = = k m + m 1 = x +

m =

t = km/n =

t

2/k = 0;_

t 1 = m'/n = 0. t' 2 = 0.

IIA. MAIN SUBGROUPS

Weights a^, (from Table I)

i = 1 2 3 k 6 Check sum

b. =

Subgroup X]_ No.

1

2

3

is5

k

SUJTl

Observations x^ in increasing order from i = 1 to i - m x2 Xj x^ xg xg Check sum Sa-x.^ Zb.jX.j_

T = Sa^x^/k + (Zb-XjAJyp =

ITB. REMAINDER SUBGROUPS

Weights a!^ b^ (from Table I)

1 = 1 2 3 4

a. =

i

b. =

6 Check sum

Observations x.* in increasing order from i = 1 to i= m»

*L

x' Check sum Za. x. Zb.x. 6

a i x x

T' = Za^xx + (Sbix

i

)yp = +

III. ESTIMATORS

5 = t T + t'T* / l

P =

estimation of extremes

kb

Worksheet 2 - Predicted values, confidence band, efficiency, plotting positions

p y

p

PREDICTED VALUES

£

p = + y?

= Q m Q_,= Q m 1 Tar(lp)

t

2

2

682-CONFIDENCE

BAND HALF-WIDTH

QIS = Qo /n

(Q0 from

Table III)

EFFICIENCY

E- QLB

(from Tat le ni) o(Ep) = </var(£

p

) yar(C

p

)

(D (2) (3) a) (5) (6) (7) (8) (9)

t

2/k =

2 t» =

P =

Estimate of u: .36788 0 p

2

p2

P2P

2

.50 0.36651 P

2

p

2 P

2

P

2

.90 2.25037 P

?

p 2P

2

P

2

.95 2.97020 P2

p2P2

P

2

.99 it. 60016 P2

p2

P

2

P

2

.999 6.90726 P

2

p 2P2

P

2

1 - - - ypP

2 yp2? y2

P 2 ... yp

2

P

2

PLOTTING POSITIONS

Observed extremes in increasing rank from 1 to n =

Rank Observed Plotting Position Rank Observed Plotting Position Rank Observed Plotting Position

r Extreme r/(ntl) r Extreme r/(ntl) r Extreme r/(ntl)

1 n 21

2 12 22

3 13 23

it lit 2it

5 15 25

6 16 •

7 17 •

8 18 •

9

10

19

20

n

Sum

45

The assumptions upon which the method is based

are that the data in the given sample (arranged in

the order in which observed''*) may be treated as

independerit random observations all from the same

population

F(x) = exp(-e- (x " u) /P)

(In cummulative form), with constant unknown parameters

u, (3 to be estimated.

For concreteness, the rules below refer to

an actual example, worked out in Worksheets 1, 2,

and Figure consisting of the 23 maximum positive

acceleration increments observed in 23 flights of an

airplane and identified as "NACA-Langley-Sample III,"

which are listed in the column headed "Observed extremes

(+An) M in Worksheet 1. These data are assumed to be

given in the order of observation, so that under the

above assumptions this arrangement may be considered

to be a random one

,

Vf If the observations are not available in their

original order, it will first be necessary to

randomize them by use of a table of random numbers
orfj B'lB

: n y

• ; 8>

: y ;

--- : v yc dh-o&dacoionl

•fS.OldB luqoq

• •• <" xo - r.x)o

> #

.-,. ...; . ,

. ‘

. , ©v y a ttesstw o ay,

, t

' r : y 1 / y>-v ad e'j <! • ,(3

*:f -'-.a , v,1^3'3*IO-IiO§ *10%

t

X; ?. «,o F-e ••vow *©Iqraax@ tBSf$h& v.IBi-vMisoq ismiAew t-S, end io yntdeisnoo .,fl S’lugl'l Bna

sis lo aaxi • ill IS -Li: Jbewtaecto mstmmi noids^a.Is&oa

%ni vLfyM&”%9L%usCL~A0&*ii! sa Baa &m lq*Ua

I X- y iC* f 5

s

:. $'& y ; ni SsdaiX @%s ffoldM

d :-

:

•

•

- n

------ H

r riri *{ -a: d

-

- f 3 d

- ' .:yv- - iso 1.0 I O' I ’1 £ - §

ho'-'ie r- 1 an ' -D n ;/-r . \ yryvy---' --o' o.?dd & no X 3 E £ svods

. OffO rtOfeftB'X a scf oi

•i -da-id nl ex diSil .I svb Xoa ©*x/j £

od v 5i?yy;- :s n>d d toi

Ilio dl .’isfeie i^nlg.Ho

c-iac»m?a ••r:o jy'i: ‘to el dad o X o sad v;d Mtodd
46

Each rule (except Nos« 2 and 7, which are

subdivided) consists of a single paragraph and this

is followed by a detailed explanation of its use,

inserted for conveniencehfof the user. This makes

* v the list unavoidably lengthy, but the rules themselves are brief and simple to apply*

Before starting the calcul ations , it is

desirable to plot the data on special probability

paper according to the directions in rule 6 (a),

’’Graphical anal ysi s

,

rt below, in order to obtain

a crude judgement of how well the data fit the

assumed distribution.' In rearranging the data in

order of size, however, care should b'e taken not to

lose the record of the original order in which the

data were taken; because randomness will then have

to be reintroduced.

Determination of estimators — Worksheet I .

1. Enter the observations in the second

column in the>’ order in which given. The

first Column is for identification purposes
' ••• b.H"

. : o'l i

'

•

;

' ‘ x i, •-

r
47

2, Determine partition of sample size (if 7

or more, but not extremely large) and

split sample into subgroups according to

the following rules (a), (b) or (c)» If

n is extremely large, say several hundred

or more, see Appendix C»

(a) If n is an exact multiple of *> or 6,

write n = k *5 or n = k*6; if both, use

n = k*6.

(b) If n is not an exact multiple of 5 or

6, write n = k*6 + m* , m’ <6, uni ess

n = 31 > 61 , etc., i.e., one plus a

multiple of 30 *

(c) If n Is of the form 30k + 1, write it

as n ~ (30k-£) + 6 = (6k-l)5 + 6, i.e.

split n up into (6k-l) subgroups of 5

it 1 H . and a remainder" subgroup of 6*

Once k, m, m ! are determined the

blanks in Section I can be filled in.

At the same time, in Worksheet 2, the

numerical values of m, m’ should be

on bored' as subscripts in the headings

W Q

u and "Q, H for columns k and

respectively. In the worked example,
- •

' ’ *

" * • • :

; ;

'

' : .

it

’

i ; . *: <

>.

'

' ]' 41

\

*?’. :'.!**’ '

• u ?. ! a 17 (&)

;; »i ©4 1 «iw

. ds H 0

U ci , 1 ( d)

lilyL!/"*.; *

"

• ' -

Xt :-;r «

** + i?- 40£} * n m

- < . . ,.s

' • v

' ' "• •;=£•

, , 7 -li

*

' : O j Otil ffl -Z-K £c

^ •: %

v! brn is r

«-

V.. If- Vd :.! 'g« s |
L8

n= 23 = 3

* 6+5 (rule (b)), so the

data are split into 3 main subgroups of

6 and a remainder subgroup of 5*

Number of decimal places . - As a result of

considerable experimentation it is recommended'

that all computations be carried to exactly

I the number of places shown for each item on

the two worksheets*

3* Find estimators for the parameters £ , u,

ty .filling' in the blanks and following the

directions indicated in Worksheet 1, Sections

IIA, II B, III.

In Section IIA, obtain the weights

a

^

9 b^ from Table I for n = m, the size

of the main subgroups* Mark off the subgroups by any convenient means," arrange the

observations in increasing order within

each subgroup and enter horizontaly opposite

* It was found convenient here to determine the subgroup size m before entering the data in the extreme

left columns, so that the subgroups could be

plainly indicated by means of a space after every

m-th observation.

49

the proper subgroup number in Section

IIA, Obtain the two product sums

m m

2 a.x., 2 b.x. as indicated in

1=1 11

A =1

two right-hand columns and sum all

columns as shown* The two productsums evaluated for the line labelled

’"Sum'* will serve as a check, Form

the average. T, by dividing by the

number, k, of main subgroups*

The work in Section I IB is

analogous, except that the weights

i t

, b^ are the and b^ shown in

Table I for n = m* , the size of the

remainder subgroup; also, since there

is only one subgroup, averaging is

unnecessary.

Section ITT combines the (sut)estimators T and 11 w' th the proportionality

coefficients t, t

s , determined in

Section I, to produce the final overall sample estimator

= tT + t

1 T

? = *929^6 + *16774 yp

, on collecting the coefficients of y^

50

and the constant terms. The estimates

of the parameters u, p are read off

A at once from the coefficients of £

and entered. This constitutes the fitting

of an extreme-value distribution to the

given data.

Predicted values (etc.) - Worksheet 2«

4.

A

Compute the values of in column 3 for

the values of P apd y^ shown in columns

1, 2, These values constitute the set

of predictions for the respective probability levels.

Additional probability levels may

b8 inserted between those shown, if desired*

The value of y

= -log (-log P) is found

Jr 00

most conveniently from Table 2 of reference

15.

5. Confidence-band half-width (68-percent

control curves) are computed from the standard deviations as indicated

51

The numerical values of the variances

v in columns and 5 are found under

these same headings in Table III, Part

A, and entered as shown* The values of

2 2

t /k, t* are entered above these values,

as indicated, in order to facilitate computation of the variances of the overall

es timator,

V" ( h> =T QTM

+ t,2qm

'

in column 6* Column 7 gives the standard

A deviation of the estimator £ » I t is most

P

easily computed by taking the square root

2

of the coefficient of (3 in column 6 and

by

mul ti plying Athe value (3 found in Section

A III of Worksheet 1* Thus a(£p) for

P = *50 i s /*0605T times the value

(3 = »1677U (written at the top of column

7 for convenience), giving the value »0ij.l3

shown*

The standard deviation of the estimator

measures the reliabili ty, that is the extent

to which repeated application of the procedure
MV

;A

/
52

to repeated samples taken under the same

conditions would give values clustering

more or less closely about the unknown

parameter value® For example, for a fixed

probability P # in about 68 percent of the

time (when the assumptions are satisfied)

the ' computed interval £

plus or minus one

standard deviation will contain the true

unknown parameter^ ^ = u + pjp® For two

standard deviations the percentage rises to

95" '^wo curved lines, one joining the lefthand endpoints of these intervals and one

joining the right-hand endpoints, are called

control curves (see rule 7 for graphical

analysis, below) and these two curves define

a confidence band consisting of the area between ' them. The interval of values of the

abscissa x = ^ included between the

These percentages are only approximate since they

assume £p to be normally distributed. As indicated

in Appendix D, this assumption is sufficiently

correct for practical purposes for samples of the

order of 100 or more. This may, of course, not be

the case for much smaller samples* However, normality

assumptions of this kind must often be made in

practice in the absence of large-scale investigations

"to establish more precise distributions® Results

obtained in this manner have often been found to be

satisfac tory«
.

..

'

• ,
53

control curves, when P is given a specific

value, is called a confidence interval „

The standard deviation in Column 7 is

thus the half-width of a 68-percent confidence band (or interval)* If levels of

95 percent, etc*, ar-e desired, the values

can be readily obtained by adding another

column consisting of twice (etc*) the entries

in column 7*

6> Efficiency is computed as follows* The

values of Q for the indicated values of

o

P are taken from the column headed in

Table III, Part A, divided by the given

sample size n, and entered in the column,

8, of Worksheet 2* The efficiency is

obtained by dividing this by the correspond

2

ing entry in column 7? canceling the (3 (which was one reason for carrying it along

separately) , and finally entering the result

in column 9®

7 » Graphical analysis consists of plotting

the data on suitably ruled paper, drawing the estimated straight line, drawing .

54

in the control curves, and seeing how well

the data fall within them. The method is

essentially due to Gumbel (c£« -reference

5 ) •

(a) In the section of Worksheet 2 called

Plotting Positions , arrange all n obser-

/ vations in the sample in a single ascending series from smallest to largest

and enter them opposite the rank nun®

bers r = 1 to n„ Compute and enter

the plotting positions fj)(x) = r •

• n + 1

Then, on a sheet of Extreme Probability

Paper"

11 such as used In Figures 4 and

5, plot the points (xr , The

observation x^ is plotted on the uniform

scale along the horizontal axis? the

fraction is plotted along the

nonuniform vertical scale <{)(x)o These

points are plotted as shown in Figure

4 *

# I.e., coordinate paper with one scale (x) uniformly

spaced and the other (y) distorted In such manner

that the extreme -value distribution exp(-e~y) will

. plot as a straight line*
• v . . ;

•

* . y

•

. .*-
55

(b) After the points are plotted the

estimated line ‘X - u + 3y, i.e„,

x = ,9295 + .1677y (see Rule 3,

above), is drawn tkrough them* This

is easily done from columns 2 and 3

(Worksheet 2), since column 3 gives

A

the predicted values of x(=£p) corresponding to the values of y(=yp)

in column 2, An even simpler method

is to take two or three widely separated

values P in column 1 together with the

A

corresponding values £ p

, plot them on

the <j)(x) and x scales, respectively,

and draw the line through them*

(c) The 68-percent control curves are

obtained by measuring off horizontally,

at each value of P in column 1, the

A

distance a(£ ), taken from column 7, to

the right and left of the fitted line,

and then joining all the right and

i'

’ all the left endpoints of the intervals

so formed, as in Figure Ij. The area

included between the two control curves

56

is the 68-percent confidence band*

If most -or all of the plotted points

fall wi thin the band, as in Figure ii,

then we conclude that the fit is satisfactory and furnishes no evidence

that any of the basic assumptions are

violated.

(d) The fitted straight line provides the

'•

i predictions for any desired probability

level P," For example, the prediction

for P = .995? which means a value of

acceleration increment which has only

one chance in 200 of being exceeded,

is obtained (in Figure 4) by reading

across to the solid (fitted) line at

P = .995? and down to find the value

i x = l

, 0 82g« This Is sufficiently close

to the value 1.8176 obtained by calculation, using the value y^^ = 5,29531.

The 68-percent curves give a confidence

interval for this value of approximately

1.66 to 1*98. This means that there

On the probability paper (Figures k?5)? P is

denoted by |T(x)
• \ „ ‘ ' '
57

is a probability of about two- thirds

that such' an interval includes the

true predicted value at we are

w P

trying to estimate* The efficiency

associated with this estimated is

between 80,3 percent and 82,6 percent

(column 9) ,

••slUff':ic;iently narrow limits

for practical purposes, r f a more

accurate, value for the prediction

or measure of efficiency is desired,

it can be readily obtained by inserting a

M P = *995 line” in the first

table on Worksheet 2 and performing

the computations indicated in columns

2 through 9®
. c 

" : " ' •
6, COMPARISON WITH METHOD IN PRESENT USE

It Is of Interest to compare the proposed

order-s tatistlcs method with the method of moments

of E, J. Gumbel which has been used up to now In

extreme gust-load computations (reference 18) • The

comparison is presented in two aspects — theoretical.

Involving an empirical attempt to evaluate the bias

and effi ciency'k of the Grumbel estimator; and practical,

showing how the two methods work out in an actual

example

6oX Theoretical Comparison

Only the general results will be indicated

here, the details being furnished in Appendix E.

The comparison consists in writing down the Grumbel

estimator, a function of the observations involving

the sample mean, standard deviation, and the profeability factor Jp,. and then obtaining the bias and

the relative efficiency of the proposed order-statistics

estimator to the Gumbel estimator,,

# For theoretical comparison of confidence bands,

see Appendix D„
.

f. .
59

Of the two characteristics bias and efficiency*

the main interest at this point is in determining the

efficiency of the proposed method, since that is the

important feature whereby possibilities of cost

savings, through taking fewer observations 5 can arise

,

Bias is less important for this purpose, and its

consideration is therefore limited to the appendix

(E),

As shown in Appendix E, (Section 1 relative

efficiency involves the first two moments of the

sample mean and sample standard deviation, and the

covariance of the mean and standard deviation. Of

these, only the first two moments of the sample

mean can be obtained readily by standard procedures,

while the remaining three quantities would require

a prohibitive amount of numerical integration to

evaluate accurately.

^T'l’he present 'discussion compares the order-statistics

estimator with the Gumbel estimator i ~ x +

,/7~

. Cr #(yp

-/)sx * As explained in Appendix E, this estimator

is r

a simplified form of Gumbel 1 s original estimator,

and is used when the sample of extremes is large.

Appendix E also considers the original Gumbel

estimator, which is a more complicated expression

used for small samples, and shows that this estimator

is both more biased and much less efficient than

the simplified estimator.
.

• • •; , V;>.:

-•

. : • ... i.

'

!

’

' * - ;

;
60

Resort was therefore had to a method whereby

the theoretical extreme -value distribution was represented by a large set of suitably constructed random

numbers* By means of these numbers a large number

of actual random samples were drawn and the results

tabulated* This was carried out mechanically with

high-speed IBM equipment* By using 12,000 random

numbers, 1200 random samples of 10 were drawn and a

single average figure for relative efficiency was

computed for each set of 100 samples* All these

computations were made for the single probability

level P = Other values of P are considered

below®

The results are shown in Table V and portrayed

in Figure 6* For samples of 10, the efficiency was

greater for the proposed order-statistics estimator

in 5 rases out of 12 (relative efficiency R (column

8) greater than 1) and greater for the present moment

estimator in 7 cases out of 12® The average of all

12 relative efficiencies was very nearly unity, These

results suggest that, for samples of 10, the two

methods are equally efficient.
2J

v - • ..

:: • . ..
61

The entire procedure was repeated for samples

of 20, obtaining 6 (instead of the previous 12) values

for the 6 sets of 100 samples each. As Table V (column

9 ) and Figure 6 show, the balance now was £ to 1 in

favor of the proposed method, with the average being

1.11, representing 11 percent greater average

efficiency for the proposed method.

For samples of 30, there were 1| sets of 100

samples each, and the results (column 10) were 3

to 1 in favor of the proposed method. The average

relative efficiency was 1 . 13 , representing a 13

percent gain in average efficiency.

To see the effect of different probability

levels on these results, computations were undertaken for several values of P beyond *95 • However,

in order to avoid needless calculation, in view of

the fact that only qualitative conclusions are

warranted, the above procedure was modified as

follows. The sets of 100 samples were combined for

each sample sise, and a single overall average for

relative efficiency was obtained for the 1200 samples

of 30, the computations being carried out for the

selected probabilities P = . 95 ,» 99 > * 999 , and the
/ .4. i

:

:: --:r -uv\

. ... r
62

limiting value, unity* The results are shown in

Table VI. In addition,- theoretical calculations**

were made to obtain the asymptotic relative efficiencies

as sample size increases without limit* These values

will be found at the bottom of column 9 of Table VI,

The above add! tional results indicate that

increasing the probability P tends to increase the

efficiency of the proposed method relative to GurabeX ' s

,

It should be pointed out that these values

obtained from the empirical sampling method are

indicative, rather than conclusive, on account of

the random variation inherent in the method, as manifest in the wide fluctuation in efficiencies shown

in Table V for the individual sets of 100 samples.

Nevertheless, the above results do give strong indication for the following statement:

For samples of 10, the proposed orderstatistics method is about as efficient

as the method of Gumbel, while for

samples of 20 or 30 or more, the prooosed method is more efficient.

Since these calculations are mainly of theoretical

interest, they have been omitted in order to keep

this report from becoming unduly long.
i

• *j
63

For ? = *95 or greater, this Increase in

efficiency is about 12 to 15 percent

for samples of 28 to 30, and ultimately

rises to 25 to 30 percent for indefinitely

large samples

If, in the comparison presented above, the

simplified Gumbel estimator is replaced by the

original form of the estimator (Appendix E, Section

2), then the comparison becomes much more favorable

to the proposed order-statistics method and it can

be stated that:

For samples of 10, 20, and 30, and

? = .95 or more, the order-statistics

method is up to twice as efficient as

the Gumbel method using the original

estimator. Moreover, this 100 percent difference in efficiency between

the two methods is of sufficient magnitude not to be significantly affected

by the sampling errors inherent in the

method of evaluation.

6*2 Comparison based on a sample of actual observations

We shall use the same data already analyzed

by the order-statistics method, consisting of the

23 maximum acceleration increments listed in Worksheet 1. For convenience we shall use a standard

form of worksheet, employed by the Environmental
V -

v'

-

•: - ! v

-
64

Protection Section of the Office of the Quarterm&s ter

Genral, Department of the Army, (reference i 7 ) for applying the method of moments of E, J» Gumbel,

To avoid confusion with the Worksheets 1 and 2 discussed previously, these new worksheets shall be

referred to as Table VII—' Part A and Part 3, The

items are filled in on both parts as directed, except

that the factor N/^N-l) is ignored in Sections I

and IV, since subsequent theoretical investigation

has shown its use to be incorrect; also, the values

x^q and x^

Q^ in Section III, and the entire Section

V are not needed for our purposes. The values of

on

and yn In Section II are taken from a table

supplied with the worksheets but omitted here.

Comparison is best shown graphically, as in

Figure 5>. It will be seen that in this particular

case the fitted lines given by the two methods are

not greatly different, the predicted values diferIng by amounts varying from .03 g at the P ~ ,95>

level (1 chance in 20 of being exceeded) to nearly

.10 g for P = .999 (1 chance in 1,000 of being exceeded).

The most striking and significant feature about

the coroparisD n in Figure £ is the narrowness
& : m insists

^

Ho'i

’ ; • 4i'4v& c4

; ; ;^v,- = Vi ^:

, ?

'•• rv *9*s*r'? I-:-.; 4 ' • :-;r.

:

- J - . 4 *\ 1 Ooi.

>>-*•'« - v- ". .;• - Of

J:

* ” ' ; t-em :.:o

' •' • ^ *-A3 ...idr h;.

- rv...

. L,-.0 S,£w $ X «•? 6

ii;{/gisi

* a’SBD

v*l ?

: ,:. .

r -'.:w f

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65

of the confidence band for the order-statistics method

compared with that of the Grumbel mthod. This is

attributable mainly to that fact that in the case

of the order-statistics estimator the confidenceband width is based on the standard deviation of

the estimator, computed by the methods indicated in

this report, whereas in the case of the moment (Gumbel)

estimator, the standard deviation, whose value is not

known, is replaced by a standard deviation that can

be readily calculated, but which results in an unnecessarily wide confidence band (for details see

Appendix D)

„

6,3 Advantages and Limitations

of Proposed Method

From the discussion given herein it appears

that the proposed order-statistics method offers

the following advantages over the method of moments

now in use:

a. The proposed method provides for the

first time an estimator known to be

unbiased, whose efficiency can be simply

and accurately evaluated.
;

•• Vi •*> i

•- .v- - . -C .

'
66

b. The new estimator is more efficient

than a simplified form of . the Gumbel

estimator, for samples of about 20

or more and P ~ 221^ more

Compared with the original form of

the Gumbel estimator, the new

estimator is up to twice as efficient

for the same range of values of P

and for samples of 10 or more

c. The calculations necessary for the

proposed method are simple and unified, giving simultaneously (i) estimates

of both parameters, (ii) the predicted

values corresponding to assigned probabilities and the reliability of these

values, and (iii) estimates of the

efficiency of the method,

d. The proposed method uses a more

valid procedure for obtaining the reliability of predicted values, and

this procedure yields smaller

confidence intervals in many cases,

(See Appendix D#)

The following two limitations of the proposed

method should be kept in minds

a. As is true of any other method of

analyzing data, its use is appropriate

only when the assumptions upon which

it is based may be considered to be

approximately satisfied, namely?

all the observations constitute an

independent random sample from the

same population P(x) = exp (e“ (x-u)/p)

(in cumulative form)®

b« The assumption that the data are ’to

be available in the order in which

observed is of some importance.

For if the data are first rearranged,

grouped or processed in any manner.
3

4io? < o s -nets korti &v.i bev- o<'' f;y-.e adT ,b

'<w &ft$ sn in x£ r do io’i fj'iifbeooiq bx..u>v

b/ifl j [B v :j • ; s to ; a Lt Id 5 £1

£ X I&in y,

! • b .1 67 %

Q ' l SJ .C 6 0 O T ] 8 1 d3

i'ica bo van,:'.' r ': -- v ; ;on I oo '• Ktlnoo

'

- yc >

..."

... = tvroliol exii

1,/orlM ronXem

• ;

.. .i '5.4 »b

-i ' ; 5,-y; Ji-rts
67

their randomness must be considered lost. In order to use the

proposed method it will then be

necessary to restore randomness

by use of a table of random numbers to rearrange the data. This is

less desirable and the original

order should therefore be preserved

if possible.

This necessity of avoiding preliminary

processing imposes a disadvantage on the proposed

method, as compared with the Gumbel method of

moments, when the sample is very large (several bundred or more, say). In the latter method the data

may be groupie1 , simplifying the computations. The

method of order statistics, on the other hand, is

not applicable with grouped data - each observation

must be treated on an individual basis - and hence

is not suitable for occasional enormous samples,

as is the Gumbel method. However, for such masses

of data an even simpler method, described in Appendix C, is available.
. *

9-l

-

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•' * .

• . i

:k

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: • a.: •.

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-
7* CONCLUDING REMARKS

This report has developed and illustrated a

new method of analyzing extreme -value data based on

order statistics that is simple to use and offers

certain important advantages over the method of

moments of Gumbel now in use, as well as being subject

to certain limitations (see Section 6*3),

In view of these considerations, this new

method is recommended for practical use in place

of the present method of estimation.

In developing an estimator intended to be

useful and efficient a number of subsidiary questions

were encountered and treated* The most important

of these were (i) obtaining mini mum-vari ance unbiased

linear functions of order statistics for small samples,

and (ii) finding the most feasible v/ay of breaking

up a large sample into subgroups small enough to

take advantage of the results in (i). In addition,

considerable attention was given to a number of

theoretical points of difference between the present and proposed methods.

Such theoretical study showed that one

feature of the present Gumbel method, namely determination of the confidence intervals or control curves
; y ;;; • •••* S.:';-.'

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69

for large values of P, does not have a suitable

theoretical basis., and that certain adjustments

should be made in the formulas,, These adjustments would have the effect of replacing the parallel

control lines by diverging curves in the regions

of high values of P, resulting in smaller confidence

intervals for the more common values of P and larger

intervals for the higher values of P that occur

less often in practice.

The solutions to the above two main auxiliary

problems have been incorporated into a simple set

of tables and a pair of unified worksheets designed

so that the computations show at a glance the essential

quantities of interest - the actual predictions, their

reliability, and the efficiency of the methocU The

method includes provision for showing these results

graphic ally.

The present study has also devoted some attention to a method involving empirical random

sampling and IBM tabulating equipment in cases

where direct numerical evaluation is prohibitive.

The use of 12,000 random numbers and from 400 to

1200 random samples was found insufficient to yield
<•* - 2.

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' "• sm .

- gffe

j

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70

accurate quantitative results for one form of the

Gumbel estimator (the simplified form) on account of

sampling variation. However, definite qualitative

results in favor of the proposed method were indicated

in the case of samples of 20 and 30 and theoretical

calculation, showed that this advantage was considerably greater for indefinitely large samples.

As a result of the experience gained in these

studies it seems likely that for accurate results

perhaps ten times the number of samples used (or

more) should be taken and the computations performed

through specialized procedures on high-speed electronic

computing equipment.

Further calculation showed that in the case

of the original form of the Gumbel estimator,

much more definite statements were possible concerning

efficiency. In this comprison the proposed estimator

turned out to be up to twice as efficient as that of

Gumbel, not only for the sample sizes 20 and 30, but

down to samples of 10 as well. For very large samples

this advantage dropped somewhat, but the proposed

estimator remained at least 20 to 30 percent more

efficient.
} f "

; -v ,

'

.v i'

' • V -V' ;

f.

'

' T ,

: - \:

f V

/<* ii’iVCJ

.

. v f

t blK
71

ACKNOWLEDGE!®N TS

The author has benefited greatly from the generous

cooperation of a number of persons in connection with

the work embodied in the present report* He wishes

particularly to stress Ms gratitude to those whose painstaking efforts'1 were indispensable to the successful completion of the entire projects Dan Teichroew of the

University of North Carolina* for carrying out on IBM

equipment the empirical sampling procedures wich made

possible the comparison of methods in Section 6,1 and

Appendix E; Irene Stegun* Computation Laboratory* National

Bureau of Standards* under whose supervision the basic

computations described in Appendix B were performed; and

to the:* following personnel of the Statistical Engineering

Laboratory* National Bureau of Standards — I. R* Savage

for contributing Appendix A* concerning the non-existence of

sufficient statistics; Yvette Bernardon* for her patience

and efficiency in typing the stencils for the entire text

and appendices; L, S , Deming* for her success in producing

the particularly effective form of the tables and figures

from unusually difficult material; and Shirley Schleter,

for valuable typing assistance In the draft stages of the

manuscript. Thanks? are also due to V/. R, Knight, guest

worker from Antioch College* for the very numerous smaller

computations *
B-t

'

v :d at,'

: . V,
REFERENCES

1. Cramer, Harald. : Mathematical Methods of Statistics e Princeton University Press,

Princeton, N« J. , 1946

2. Gumbel , Emil J. : Les Valeurs Extremes des Distributions S tat is tiques « Jtnnales de

l’Institut Henri Poincare* vol * 4# pt. 2,

1935, PP. 115-158,

3 . Gumbel, Emil J. : The Return Period of Flood

Flows* Annals of Mathematical Statistics,

vol. 12, no* 2, June 1941, PP* 163-190,

4* Gumbel, Emil J. : On Serial Numbers* Annals of

Mathematical Statistics, vol. 14, no* 2,

June 1943, pp. 163-178*

5* Gumbel, Emil J .

The Statistical Forecast of

Floods » The Graduate Faculty of Political

and Social Science, The New School for

Social Research, New York, December 1948*

6, Gumbel, Emil J. t The Statistical Theory of

Extreme Values and Some Practical Applications* (In process of publication by

the National Bureau of Standards).

?• Johnson, N* L, s Estimators of the Probability

of the Zero Glass in Poisson and Certain

Related Populations, Annals of Mathematical Statistics, vol. 22, no* 1,

March 1951 , pp* 94~101

8* Kendall, M. G . ? The Advanced Theory of Statistics*

Vol* .1 and II, Charles Griffin and Co.,

Ltd o, London, 1948

9« Kimball, Bradford F.: Sufficient Statistical

Estimation Functions for the Parameters

of the Distribution of Maximum Values.

Annals of Mathematical Statistics, vol.

17, No. 3 <• September 1946, pp. 299-309
•••

• - . ,, .-..i

•

:

•

'

'
73

10* Kimball * Bradford F s ? An Approximation to

the Sampling Variance of an Estimated

Maximum Value -of a Given Frequency Based

on Fit of Doubly Exponential Distribution

of Maximum Values* Ann els of Mathematical

Statistics , vol. 20, no. 1, March 1949*

pp. 110- 113 .

11* Lehman, E, H.: Notes on the Theory of

Estimation, Ch. IT, pp. 1.9-19* (Unpublished

mimeographed notes of a series of lectures

given at the University of California in

1949-50)

12. Lieblein, Julius 2 On the Exact Evaluation of

the Variances and Covariances of Order

Statistics in Samples from the ExtremeValue Distribution. (Accepted for publication in the Annals of Mathematical

Statistics )

13. Hosteller, F. : On Some Useful “inefficient"

Statistics, Annals of Mathematical Statistics,

. vol. 17, no, 2, December 1940, pp. 377-408.

14* National Bureau of Standards t Table of the

First Moment of Ranked Extremes, NBS Report

1167, September 20, 1951® A special report

submitted to the National Advisory Committee

for Aeronautics.

15® National Bureau of Standards? Probability

Tables for the Analysis of Extreme-Value

Data, Applied Mathematics Series, No. 22.

(In press )

,

16, Newman, F. W.J The Higher Trigonometry!

Supperrationals of Second Order. Macmillan

and Bowes, Cambridge, 1892, pp. 64-65

17® Office of the Quartermaster General, Department

of the Army; Evaluation of Climatic Extremes,

Environmental Protection Section Report

No, 175, Washington, D. C., March 1951*

Til18* Press , Harry? The Application of' the Statistical Theory of Extreme Values to

Gust-Load Problems* NACA T3ST 1926* Washington, D, C., 1949 ®

19® Smith* J» H, and Jones* H» L .

The Weighted

Mean of Random Observations Arranged in

Order of Size® (Unpublished paper)*

20. Wilks, S. S,; Order Statistics, Bulletin of

the American Mathematical Society, vol*

January 1948 » PP* 6-50*

APPENDIX A

PROOF THAT SUFFICIENT STATISTICS DO NOT EXIST

FOR THE PARAMETERS OF THE ^ EX TREME-VALUE DI S TRI BUTT ON

(Relates to Section ij.,2 of text)

Problem ; We have a sample of n from the

extreme-value population whose density function

is

a (x^u)

-SHf

a (x~u) f(x) = ae

(3 = 1/a > 0 and u are unknown and we wish to find

sufficient statistics for them.

Theory ; (1) If t = (t

f ,, 9 .,tk

) is sufficient

(i.e., is a set of jointly sufficient statistics)

>r 9

x = (xfor - (9~,...,9tm ) then the density function of

1 1*1

$

a © o $ x ) may written in the form

P(x,9) = f (t,9) g(x)

( 2 ) If t(x) = t(x 8 ) for sample points x and

x ’ , then

P (x,9) _ g (x

PTF^T©! gTScTJ - h(x,x 1 ) e

This Appe ndIx'lias'lTe e n prepared by I* Richard Savag

of the Statistical Engineering Laboratory, National

Bureau of Standards.

For convenience the symbol a is used in place of

the parameter (1/p) of the text.
. -V

V ^. \V
76

(3) Hence for all those points where t(x) has

a constant value the ratio A is free of 9, and thus

we can find sufficient statistics by seeing for

which point sets A is constant.

(ij.) Evidently* if 0 = g(9 f ) G.

X

i y = (9^, . . . , 9^) , i = l,**.,k) is a non-singular transformation of the parameters, then also

P(x,9» ) = f (t,9» ) g(x ) PTxt79tT f(t,9«) glx)

h (x,x ! ) *

using the same (set of estimators) t as for 0. In

other words, if a set of statistics t is sufficient

for a set of parameters 0, the same set t is sufficient

for any other set © J obtained from 0 by a non-singular

transformation*

Results ; We will now apply the above theory

to the problem at hand and shovr that the largest point

set on which A is constant contains nl points, that is,

it takes n functions to describe t, so that the

resulting sufficient statistic Is the trivial set,

2 n

t = (x , ..*,x ) or In other xjords,

the only sufficient statistics are the n observations

themselves, so that we do not have a basis upon which

to construct optimum estimators,,
t

-

-

•’ i

’

'
77

Analysis ; For the distribution f(x) we have

a f r ? v S ?

r -a(x.-u) -a(xj ~u)ll A = exp < a ( 2 x» - 2 x. ) - 2 (e 1 -e i J

l 1 1=1 1 i=l J. 1

i=l

If A is free of the parameters (3 and u then it is

also free of a = 1/6 and u, and so are log A and

6^ log

A

. Hence

8 a*

, , , - - v _ r -a(x^-u) --a(x, -u

)

log A = na(x“X' )- 2 [ e

1 -e x n

2

i=l

]

Let u approach - go «, We first find that x - x * in

order to have log A free of u and a. Next,

6

k

logA , , x k+1 Sr/ ~a(x «u) « k -afx^-u) %- = (-1) 2 [(x,~u) e

1 - (xi-u) e

1

6a* i=l

0 , k >« #|

and this is true for k = 1 as we 11, since x = x».

Since this is an Identity in u let us set

u = 0c We then get

n .... n

" x . e 1=1 1

ax- ;; i k - ax

.

= l x. e i

i=l 1

These are finite suns; and therefore, since

they are identitites in a, it is clear, since a may converge to zero, that

k «k

2x> 2x.

1 _ i

n n

78

Thus the largest set of points of constancy of A

are those points which give the same sample moments,

and this fact implies the desired result®

Note c Statement ( 4 ) above implies that the

result also holds if the parameter u is replaced by

£

p = u + (3y

p = u + y

p

/a®

Example ; To show how this method works for

a familiar problem consider a sample of n from a

normal distribution! here

-l/2a2

[X(x, -G) 2 - 2 (x! “0

)

2

] , A - e 1 ±

=21ogA = [2 (x^»x^

2

) - 2©2(x. -x^ ) ]/a2

, and clearly the necessary and sufficient condition

? 2 for A to be constant for all a and © is that -

>2 t 2x. 2x^ = 2x^ , which is the classical result that

the first two moments are sufficient statistics®
'

,

..

, : ; ;

-« '

: ; b

:
APPENDIX B

MATHEMATICAL FORMULA HON AND SOLUTION OF

MINIMUM-VARIANCE PROBLEM

(Relates to Section Ij. «. 3 of text)

rv We consider an estimator of = u + (3y of

the form

n

2 WjX. 9 i=l J' (B.l)

where x_ < x0 < . < x are the n order statistics

of a sample of n from the extreme -value distribution

(4*1)* and seek to find the w^ which minimize Var(L)

subject to

E(L) = |p

• (B,2)

The estimator L in (B. 8) below with weights so determined is called the minimum-vari. an c e , unbi ased , (linear) order-statistics estimator for sample size

n 0 Wri ting

x = u + (3y , (B,3)

where y is the reduced variable corresponding to

x, we also have

X

1 — u + pyj

. (B.4)

where y < yP < ... < y are the n order statistics

of a sample of size n from the reduced distribution

80

exp (-e~^

r

) 9 free of parameters. From the above it

follows that

E(x

i

) = u + pE(y

i

) , (B. 5 ) since u and (3 , though unknown , are constants not

subject to sampling variation when the operation of

expectation is performed® The values E(y^) have been

tabulated in reference ll| for i = n(l )niin (1 ,n-25>) t n = 1(1)10(5)60(10)100*.

These results give readily

E (L ) = Swi (u+pEyi

) = g p = u + (3 yp

. (B.6)

This is required to be an identity for all values

of the parameters u, Equating their coefficients

gives the two conditions on the weights : n

2 w, = 1

i=l 1

(B. 7 ) n

2 (Ey )w = y

i=l 1 1 p

where Ey^ are the numerical values tabulated in

reference II4.

-Ve The notation in the table cited differs from that

used here: E(y.

)

in this report corresponds to

E(yJ . ) in the table® n-i
I. '
81

Turning to the variance, we have

n

2 w.w . n 2 2 n n

Var (L ) = 2 w. a

' + • 2 2 w,w

i=l 1 .1=1 i=l 1 J ~x,x

i J

From (Boii.) and the properties of the variances and

covariances of linear estimators we have

2 _ 2 2

'Xj P o

* = 6 2 a? } y*

ox-x . 1

J p

2

°y

1 y

j = p2

°ij '

making an obvious simplification in notation, whence

V

R = Var(L) = (2o^w?+22 1 jW^w

j )p

2

= minimum subject to (B.7)® (3.8)

This is a constrained minimum problem for

variation in the unknown w^, and is equivalent to

finding the (uncons trained) minimum of

”

G

1 = ^ 2ai wi +S2?ci j

wi w j?p

2 + ^

1

(2w

i -l) +

^i

1 [2(Eyi )wi -yp

],

where jiu are the Lagrange multipliers. Since

2

(3 > 0 is constant, though unknown, this is the same

as minimizing

G^ ? 2

G — ki w 22^ o

^ w

j ^ v Sw^ X )

^i[2(Ey

i

)v;i »yp J , WYhe~"tempor ary notation yu,

fjshould not be confused

with the symbols for moments.
rn

v

:

'

-

.
82

2 2

whereas ?[ /$ » f1 ~ p

* Setting the derivatives

with respect to w, , k = 1, 2, , n, equal to 0

K

and dividing by 2, we have

2 n . °k

W

k

+ °ikW

i

+ '

/

' +yuE7j

c = 0, k = l,2,»,,^n (B,9)

i~l

(i+k)

These latter are n linear equations which, with the

two in (B.7), form a simultaneous system of (n+2)

equations in the (nd-2) unknowns w^, W£, . w ,

p

6 The values of X and

p

are useful as a check, since

if (B.9) is multiplied by w^ and summed, the result,

in view of (B.?) which the wr’s satisfy and (£ 0 8) is

V . + yf+ /UY— n,mn /v

/ J ?

that is, we should have

0

Vn,min 7

The minimum value V„ » „ x-d.ll be denoted bv Q, » Before solving the set (3„7)*> (B.9) it is

necessary to determine the coefficients in these

linear equations. The values of Ey^f are tabulated,

as already mentioned. The variances and covariances

2

Ok, involve complicated integrals. The author

has been successful in expressing these integrals

83

in terms of simpler ones already tabulated (reference

12), The results ar-e, for the variances

of " E(yf) » (Ey

i

) 2

j p 00 i ^

jj E

<yf> =TTrifn^T)T

rf0

( - 1)c rg

2

(i+I,) »

1 = 1 .2 8 . • t $• * 9 f n

where

g^(i+r) = f + (/+logi+r ) ]

and { = Euler *s constant = ,57721 5&61j.9«»®J and for

the covariances^

= E(yt yj) - (Eyi

) (Eyj)

E(yi yi

} s

IT-17 TT 3^1-I )T( jTT

j-i-1

2

r=0 s=0

°C

jB ‘“ i “ 1

C

n'"%(i+r, j-i -r-J-s),

r s

i < j j > *j -• 1 J «J J e » , s ^ y

where the function 0 is defined by

2tu0(t,u) = (u-t)g

2 (t+u) + t'*[g1 (t)] ~ 2L(1+^-) + ,

in which g2 is the same function as before,

g (t) = |(3

/+logt) , and

All 1 og ar i thms are natural logarithms, to the base e
-

_
84

3 +x

L (1+x) = _(

' dw

CO t - 1 ) n+1

x

n

n

2 2 = Idog x) + ^- l ^ +x)

is Spence ? s integral, which has re^r: lost extensively

tabulated (to 12 places) in reference 16, The function

g^ also occurs in an expression for the means:

E(yP

= irnriTn-oTT 2 ( -u

r

c n 1

s.

l o.+r*} . The above formulas have been evaluated as far

as n = 6 and the results are listed in Table II,

The values in the table are believed to be accurate

to the number of places shown. Those for the means

agree (to within a unit in the 7^ pi see) to the 7

places to which the means have previously been tabulated..

Table II thus provides the coefficients in

the system of equations (B*7)> (B»9) in the w and/1 ,

jUc The right-hand sides of these (n+2) equations are

1, Yv>* * . » , 0 and the solutions w. , /v> jx are

linear combinations of these with numerical coefficients

2

which involve only o.

? , a, and Ey^, but not y^.

Hence the solutions are all of the form

w

i = y

+ hyp

* A = ox + d

x yp

p

= °2 + d

2 yP

•

i = 1, 2, n
>

-

.

; ;

•:

.

_
Substituting these w^ in (B.8) we have an expression

of the form

<4 = V . n n,imn (An yp+ B„yD+

n-'P n

(B ,10)

The quantities h^ for the weights w , and the

coefficients A

n

, B

n , Cn

, of Qn , are given in Table

I for n = 2 to 6, The solution of the system of

equations became increasingly lengthy for increasing

values of n, with correspondingly diminishing accuracy,

so that the computations were discontinued beyond

n = 6, The procedures for handling samples larger

than n = 6 are explained in the text of this report.

APPENDIX C

SHORT-CUT METHOD FOR VERY LARGE SAMPLES

(Relates to Section of text)

If we have a sample of several hundred or

more extreme observations, as may sometimes be the

case (e.g. reference 18, where a sample of ij.85

extremes was analyzed) it is possible to select just

three out of all the observations and from them obtain

useful estimators.

This technique is based on a method used by

F. Mosteller (reference 13) for samples from the

normal distribution,. If the n -sample values from a

(continuous) population whose density is f(x) when

arranged in ascending order are denoted by the order

statistics x , x^j, •••, x^, and n Is very large, the

application of Mosteller ' s method involves taking the

observations whose ranks are^n, yun, -Jn, where

0 < /l<

p

< V< 1 with/}, ju, -j suitably determined,

and choosing a and b so that''

When [as will generally be the case) the ranks 2 n,

pn, zJn are not Integers, they will be defined to

he the nearest integers to these quantities.
" ; j:

i.

r

--- '
87

E = ax + b (x - x« ) (C *1

)

v jun i/n

is an (asymptotically) unbiased estimator of the

parameter g = u + py^* (The reason for choosing

this particular form is discussed below)®

The mean and variance of the estimator £ in

(C.l) are computed from the corresponding moments of

order statistics of the form x% , with n very large An

andX a proper fraction not too near 0 or 1* Under

these circumstances the theorem used by Hosteller

states that in the limit, as n increases indefinitely,

(i) becomes normally distributed, with mean

4 n

and variance

E(x, ) = t.

An A

2, s ? a-a). ° x

;m

- TrfrhpT2

h

where t is defined by A

A J1

(C ,2)

(C®3)

f(x)dx; and

(ii) the covariance of any two order statistics

x^

n

and x , ^ < fa, is given by

oov (X

/In* V’

= nH"t^ry) • (c * w

where t is defined similarly to t/\«

P *
V .

, r?.H /

.

:

•v

'
88

For £ in (G ,1 ) to be unbiased in the case

where f(x) is the extreme value distribution*

E (|) " Ip ~ u +

(G ®5)

must be an identity in u, (3. Vie first note that

from previous discussion in the text (see Section

4.1), the parameter | is precisely the abscissa of

P

the ordinate which cuts off the area P to the left®

Hence we have simply

= u + py^ . (c ,6)

Equations (C.l), (C.2), and (C*5>) then give

at + b(tv - t

3

) = u + py

A

1

. or

a(u + py^) + b(y^ - y^)p = u + y

p

p #

from which, on equating coefficients of u and p,

yp

- y, a = 1, b LE (C.7)

ll/ - \

In principle, the fractions A , p, if might be deterA

mined so as to minimize the variance of £ and thus

make its efficiency a maximum, but this would require

very extensive computation which would not be warranted

on account of the limited importance of efficiency

when the available sample is very large. (For example,

a 5>0 percent efficient estimator with a sample of

39

1,000 gives results equivalent to using a sample of

900 —- still a large sample,'*) Instead

we consider estimators of f of the form

t A A l = u + ypi8 , ( 0 a B

)

A A

where u, (3 are estimators of the two parameters u,

(3, that Involve the fewest possible number of order

statistics x without undue sacrifice in efficiency

sn

as computed for idenfinitely large samples*.. The

aim is to find, with a minimum amount of computation,

o A

separate unbiased estimators u, {3, of the parameters

u, (3, each of which has minimum variance or best

efficiency in some sense, in the hope that the linear

combination (0,8), i^jhich will also be unbiased, will

turn out to have efficiency which is not unreasonably

small* ‘Aiis is a heuristic method, since the fact

A a that u and B are efficient does not Imply that their

These considerations assume that the sample of

data is already at hand, perhaps by a survey already

made, such as the Thunderstorm Project mentioned in

reference 18, Of course, if it is a question of

planning for the securing of data, it is desirable

to use as efficient an estimator as possible, but

in that case, the Investigation will rarely be

sufficiently extensive to provide samples large

enough for the method described in this Appendix

to be applicable.

90

A /\ combination u 4- ypP is efficient. Much better

estimators probably exist, but we are content to

obtain just one of reasonable efficiency.

It turns out that the modal parameter u can

be estimated by a si ngl

e

order statistic. E* J,

Gumbel has shown (reference ii, equation ( 5>0 ) ) that

the value of u for which x^

n

best (i.e,, with least

variance or most efficiency) estimates u is ju - .20319®

For simplicity, we therefore replace u in (C.8) by

“ = x .20n * (C<9)

The scale parameter p requires at least two order

statistics, or rather their difference x - x „ , fc/n ^n

for estimation, multiplied by a suitable unbiasing

factor which will become absorbed in the expression

for b in (C»7)® A considerable number of trials

indicates that the pair of values } - .03, *85

gives an estimate of p with efficiency probably close

to the maximum, if not actually maximum. Since we

are not seeking very precise results, this pair of

values is adopted here. Thus (C.l), in view of

(C.7)» becomes

i

= x

. 20n

+ °.32S6(yp

+o.U7S9)(*. 8Sn

-* >03B ) . (C.XO)

The variance of this estimator is obtained from the
. .- . . . '':• .. • 5

; -,i

"" ' d
91

rule

m m 2 2 m

Var( 2 a.x.) = 2 a. a + 2 ° 2 a„a^ cov(x.ji.)»

1-1 11 1=1 1 X i i,J=l 1 J 1 J i<j

which after simplification gives

o

2

{ |) - 8o6916 - O 0O68I d + lo 51^2 , (Coll)

where

a = 0.3256 7p + 0.1549 . A

Since g

is, unbiased* a measure of its efficiency

may be obtained by dividing its variance into the

Cramer-Rao lower^feouacL Q,__ (see equation ( iq 0 1 9 ) and ho

accompanying text* numerical values are given in the

Q, column of Table IH, Part A, The results are as follows g for several values of P of interests

P Efficiency of H

79^ jSU5-

°99 Jbij.9

°999 0652

1( limiting O660

value

)

Thus , this 1 arge~ sample method of estimation is

slightly less than two- thirds efficient 0 However*

as noted above* such apparently low efficiency need

not be a serious matter in practice 0 For convenience* a summary of the method

described above is given here

92

Summary of Large -Sample Procedures

1. Arrange all n observations (assumed to be

independent and from the same extreme -value distribution) in order of increasing size* and then

rank them from 1 to n,

2 0 By hand or mechanical sorting* select the three

observations x whose ranks are the nearest inr tegers to G Q3n* o 20n* and 0 85n 0 Denote these by

x

«Q3n*

X

o20n*

x

»Q$n°

A

3* Compute the predicted values ^ * for various probability levels P* by formula (C o 10) o

lj.«. For each P cor^pute the variance from formula

(caih

5* Take the square root of the variance to obtain

the standard deviation* This gives the half-width

of the 68 “percent confidence b an d * since for

A large samples the distribution of £ approaches

normal! ty 0 Sinilarly* twice the standard deviation

determines the 95 percent confidence band„ and

2 o 58 standard deviations determines the 99 parcent bando

Go Obtain the efficiencies by dividing the variance

into the Cramer-Rao lower bound in Table III,,

Part Ao
* •

•

.

'

.
APPENDIX D

ANALYSIS OF CONFIDENCE INTERVALS IN ORDER-S TATI STICS

METHOD AND METHOD OF MOMENTS OF GUMBEL

(Related to Section 5* Rule 5? and Section 6,2 of text)

1. Confidence intervals in orderstatistics method (based on

normality assumption)*

In the text s Section 5s> Rule 5s the confidence

intervals given for various confidence levels in the

proposed method are obtained by laying off a certain

number of standard deviations 9 computed for the estimaA tor gp, on either side of the estimated value given

by the fitted line. If this Is done for different

values of F and the ends joined,, as in Figure 4s a

confidence band is obtained. The number of standard

deviations given in the method — one for a confidence

level of 68 percent , two for a level of 95 percent —

A is based on the assumption that the estimator | Is

P

normally distributed. The purpose of this section

Is to investigate this assumption more closely.

It will be recalled that the estimator ^ is

obtained by splitting the sample Into a number of

equal groups with perhaps a remainder of different

size (see text In connection with equations (4*22) a

(4.25) and (4.26)). Then 2 can be written :ion

91+

( 4 . 26 )

£ = t T + t

! -

T

s >

where T is the average of a certain linear function

of the sample variables (equation (4.22)) taken over

the k subgroups , T> is another linear function and

A

t, t

9 are constants. Thus £ is the sum of two parts:

(1) an average of k independent random variables

(tT^)'f all with the same di stribution, and (2) a

single variable (t'T 1 ) with a somewhat different

distribution. By the Central Limit Theorem in probability (reference 1, page 2l£), according to which

the average of a number of random variables having

the same distribution (with first two moments existing)

is asymptotically normal as the number of variables

increases indefinitely, the first part is approximately

normal for large k. In fact, extensive experience

has shown that a normal distribution is often a re=

markably close approximation even if the number of

variables k is under 10. Furthermore, the first two

moments (actually all) of each variable certainly

exist ~= in fact the proposed method is based upon

'These variables are independent because the subgroups

were assumed to be formed independently.

95

their computed values, ^ence, it is safe to say that

for k = 10 or more the first part is very closely

normal o The second part (t ! T 5 ) is a variable which

has the same general character as T (a weighted sum

of order statistics [equation ( 4 « 25 ) 3 ) and hence is

believed not to impair significantly the approximate

A

normal! ty of V Its influence is likely to be small,

especially if the number of other variables , k, is

large

»

For samples as large as 100, k = 16 if broken

into subgroups of 6, or k = 20 if broken into sub~

groups of 5* Since these values of k are considerably

4

?' ; larger than 10, the preceding discussion shows that it is

A

quite safe to assume normality for for samples

of 100 or more, so that the corresponding multiples

of the standard deviation given above are sufficiently

accurate in such cases. In fact, it is likely that

the normal approximation remains good for practical

purposes down to samples of 50 or 60„ becoming, of

course, worse as sample size decreases still further.

However, in the absence of knowledge about the exact

A

distribution of the order=s tatis tics estimator £

for smaller samples, the normal approximate
* 9 ’ i
96

apparently the only simple one available for de terming confidence limits. It may be noted that approximate

methods are also involved in determination of confidence

limits in the Gumbel method. This point is further

discussed in the following section.

A comparison of the confidence intervals (or

bands) for predicted extremes in the order-s tatistics

method and in the Gumbel method is of interest and is

presented in Table VIII, columns 2 and 4, < and 7s>

8 and 10. It Is to be noted that for prediction probability P not beyond .99 the method of order-stati sties

gives an appreciably narrower band for sample sizes

20 and 30 , and a significantly wider one for samples of

10. On the other hand, for values of P much beyond

.99, the order statistics band widens very rapidly, as

compared to a constant width for the band in Gumbel ! s method. However, there appears to be some question as to

the theoretical validity of the Gumbel confidence-band width

for large values of P. This point is also considered in the

following section.
.

’• 'V'
97

2 0 Confidence intervals for largest

extremes in Gumbel methodo

2 d Gumbel* s derivation of

confidence intervals

„

The purpose of this section is to inquire

into the theoretical validity of the confidence intervals (or confidence band) given for extreme predictions in Gumbel* s method®

In the Gumbel method the 68-percent confidenceinterval half-width for the largest in a sample of

n extremes and for all larger predicted values"' is s in Gumbel * s notation (Table VII s Part Section IV)

A

x,n - ^= l-une* f> = i/a, (D.X)

where (3 is the scale parameter (or rather*, an estimate of

it) of the extreme -value distribution from which the

observations are assumed to come® To obtain the

confidence Interval for a given prediction probability P > n/(n+l), the value A is added to* and

“ x j, n

subtracted from® the estimate given by Gumbel^ de~

1 noted by him by x (Table VII 9 Part B* Section III)

That is 5 for all values of P beyond n/(n+l ) g which

is the probability assigned to the largest value

in the sample s x

n

. For smaller ? 9 the confidence

interval Is given by a different method with which

we shall not be concerned inasmuch as the primary

interest is in large values of P correspond? ig to

extreme predictions.

98

and in this report by £ Gumbel 8 s (68 percent)

G

confidence interval for 'predictions beyond the largest

observed extreme x is thus given by

n

o j

t G ± l.Uilp, (D.2)

where (3 is the scale parameter (or an estimate thereof) of the extreme -value population from which the

e?£ observed extremes x have been assumed to comes

F(x) = J(y) = exp(-e^) s y= (x-u)/(3 0 (D. 3 ) The multipier l„lipl used for the 68 percent confidence

band is obtained by setting G = ,68 and solving for

y the equation ,

J(y) » J(-y) - C (Do4)

which is par ame ter -free 9 and gives y(C) ” y( 0 68 ) = 1«14073 (reference page 6) 0 Thus

y

= - 1 O l4073 to y

= ld4073 (D„ 5 ) is the interval for the reduced variate that cuts

off (or corresponds to) a central area of 068 under

the extreme-value density curve shown in Figure 2 0

The corresponding interval that cuts off the same

¥ From the theory of extreme values the distribution

of the largest of the observed value s s x

n?

in a sample of n extremes, is exactly an' extreme -value

distribution that has the same scale parameter p c

99

area under the original (^unreduced®9

) x-dis tribution

thus has width given by the values (D„5) multiplied

by the scale factor (3, since

x = u + yp

The half-width is therefore 1„ 14073 p s i 0 e op (D 0 1)«,

The following discussion indicates that this

method of obtaining confidence intervals is inaccurate

in two respects? (1) the confidence interval ie of

constant instead of increasing width for large P|

(2) the scale parameter used is inappropriate®

2,2 Constant width of

confidence interval

The above me thod‘ of Gumbel of obtaining con-

, A

fidence intervals (D„2) treats the estimator as

though it has an ex treme^value distribution with the

same shale parameter p as in the population underlying

the observed extremes (including the largest

extreme xn )o This assumption cannot be considered

strictly valid since it implies that the confidence

width remains constant for all large values of P p as

(Dol) does not involve P„ In other words, this asserts

that from a sample of 20 observations or e en 0 P for example p we can make
\. - Td; .
100

statements about events that will occur with probability 1 in a million or billion yet have the s ame

uncertainty of only A in our estimate for x as

x ^ n

for predictions about events with probability, say*

1 in 100o It does not seem reasonable that a limited

sample can tell anything at all meaningful about such

extremely rare events, let alone predict them with

the same amount of uncertainty no matter what the

probability of occurrence*

This lack of agreement with common sense indicates

£

A a that the Gumbel estimator = u + ypP cannot be treated,

for all large values of P, as if it has an extremevalue distribution with constant scale parameter*

Besides these common-sense considerations,

there is another reason why the ex treme -value distribution is not appropriate for the Gumbel estimator,

at least for large samples of data* The estimator

is a sample characteristic of the form

t

Q = * + k

p

sx »

£ D ° 6 ) where k is a constant for given values of P and n.

The appropriate distribution of such an expression

is given by a general limit theorem in probability

(reference 1, page 3^7^ to the effect that under broad
• . '•' " a' • » / .. -A;:- ... ... ~ . ... ' ; <; t, ? ' - ' : • •
101

conditions any sample characteristic based on moments

(such as £ ) is* for large values of n, approximately

normally distributed* Thus for large values of n

the Gumbel estimator (D*6) should be considered to

be approximately normal , with variance given by an

2

expression which increases as k^* Moreover s this

would yield a confidence band that diverges with

increasing P s avoiding the difficulty of the parallel

curves mentioned above*

2*3 Inappropriate scale parameter

Little is known about the exact distribution

/\ of the Gumbel estimator particularly for small

sample sizes* Yet even if It were an ex treme-value

distribution (of the form (D*3))* it would seem that

its scale parameter would not be p, but a certain

multiple of it s B , found below* This multiple may

be determined by considering the relation between

the variance of the distribution (assumed extremeA

value) of

Lr tributions

and the scale parameter (3^ of this dis2 .£ N x

2 2

° V

= T h

(D.7)
,, . * f;

. v ' >• " UY': '- ? \

:

•

:

1

; ,

.

. ;

. ... V

'

. :

'

'

• ;

;f -

-

'

'

• : •-t - :

' . : «« v S K 4 ,

,

. -

v : h "• V;

; ::ii l vif;:

'

- - -t* ;

:• .> .

'. V

'

- '

. -V

- VC?!: 1 : .

. \j .

'

'
102

But we have an approximate expression for the left

side j, namely, (E„ 6 ) in Appendix E below,, This is of

the form

°

2

(| g

) = q(y

p

) ^ (o 08 ) where p is the scale parameter of the original

( extreme “Value ) x-dis tribution, and q(yp

) is a

quadratic expression in the probability factor yp

with coefficients involving the quantities o (s) and

covfyjs), whose computation by empirical sampling is

indicated in Appendix E ; q(yp

) may be regarded as a

known value, depending on P 0 Hence

o

2

(t0

) = p

2

. (D.9)

Substituting in (D»7) gives

P = {— S

p

) p = BpP , (D.io)

1

which defines the multiple B^o Thus the confidence^

interval half<=width (D 0 1 ) must be replaced by

A 9 = 1.141 B

pp , (D 0 11)

where now A

9 is no longer constant with P, but

on account of B , actually increases very rapidly

P

for large values of y^ corresponding to values of

P near 1. Thus we obtain a modified confidence

band whose divergence states that the amount of un=

certainty increases without limit as we at temp

103

estimate increasingly improbable events. This also

avoids the conflict with common sense mentioned in

Section 2.2.

Ihe actual values of Bp are of interest and

are given in the following table for several important

values of P and for the three sample sizes for which

tLsy were computed in Appendix E.

B

P

35 S

P = ir

p

n = 10 n = 20 n = 30

95 .749 .560 .458

99 1.093 .825 .673

999 1.593 1.208 .986

In this table the values of B

p

less than 1 indicate

that the modified confidence band (equation (D.ll))

is better (i.e. narrower) than the Gunbel confidence

band and vice versa for the values of B

p greater

than 1. Thus, the modified band is indicated to be

considerably better in the region P = *95 to *99 for

samples of 20 and 30. For samples of 10, the advantage

is less at P = .95 and becomes reversed in favor of

the original Gumbel confidence band at P = .99 ,
-V" f'; -

.

' . . •

•

.

• •

• • * , : : . . <

i - ‘

’ "V .•

• » if"/’

'

: iV

!

;

' Mi: '

. v V/;

ff

: ;

£L. ::

* »*

v‘ \ •

i

' ••

’ E }

• ; •• o&. M:-

:

.

•, v

.

.

'

/’: i-f

-

: ;V h ""

• vv y- : ' 4

;

•

' ;

- v v-

f .1 ''VO ' ' ft:

•
loll

The above comparison remains exactly the same

for any other confidence level, it being merely necessary

to replace 1.1|1 in equations (D.l) and (D*ll) by the

corresponding value y(C ) determined from (D„l|)« Thus,

for the 95 percent level, y(,95) = 3»06685 (reference

6, Lecture 3> Table 3«l)o At each level the confidence intervals of the two methods are affected in

the same ratio by such multipliers, that is, their

ratio to each other remains Bp, regardless of confidence

level C*

3 » Comparison of confidence intervals in

the Gumbel method and the

method of order statistics

Table VIII shows the actual confidence intervals (in terms of the scale parameter (3) for the two

levels C = ,68 and C = *95 for the Gumbel method and

as modified by the factor Bp, and also compares these

(where applicable) with the intervals given by the

order-statistics method. Except for samples of 10,

for which the Gumbel interval is apt to be narrower,

the modification denoted by B , discussed in the

P

previous section, reduces the interval-width for P = ,99

(and less) by significant amounts — by about onesixth or more for samples of 20 (columns 8) and by
'

.

' 1
105

about one-third or more for samples of 30 (columns 8,

9). These results are of course implied by the values

of Bp., given in the preceding section* Also, as

noted in Section 1 above, the order-s tatistics confidence interval is narrower than the (unmodified)

Gumbel interval in many cases, for P not beyond .99*

and sample sizer not below 20. However, it increases

beyond the constant Gumbel width for larger probabilities, in- agreement with theoretical requirements.

At P = .99 or less, there are two additional features

to be noted. (1) With increasing confidence level,

numerical

the/factor in the Gumbel interval A increases

x p n

faster in either the modified interval A

8 or in the

order-statistics interval (denoted by A in Table

VIII ), so that both the modified method and the orderstatistics method reduce the confidence interval of

the Gumbel method by constantly Increasing percentages as the confidence level increases. For example,

for P = .99 and for samples of 20 the order-statistics

interval is about 11 percent narrower than the Gumbel

t interval for a confidence level of 68 percent and

about 30 percent narrower for a level of 95 percent

(columns 5j> 7)® (2) Similarly, the percentage
\

'
106

reduction increases with sample size. Thus* for

P = .99 ana a confidence level of 68 percent* the

reductions are 11 percent for samples of 20* and 29

percent for samples of 30 (columns 8* 10)*

APPENDIX E

DETAILS OF THEORETICAL COMPARISON BETWEEN ORDERSTATISTICS ESTIMATOR AND MOMENT ESTIMATOR OF GUMBEL

(Related to Section 6.1 of text)

Since the order-statistics estimator has been

fully discussed in the text, the remaining problem

in making the above comparison is, essentially, to

develop the characteristics of the Gumbel estimator*

The method of moments of Gumbel In present

use provides the following estimators for the parameters u, p (reference 18, page 11, equations (26),

( 27 )? also reference 5> page 10* equation ( 29 ), but

read (

»

yn/a ) for (+yn/a)) :

where x, s are the mean and standard deviation of A

the given sample of size n; y^ is a certain computed

quantity, depending on the sample size n, which

approaches Euler’s constant Y - .577^0.® from below

as n becomes infinite; and a is another computed

quantity, depending on n, which approaches %/ </E = 1.28255®°° from below as n becomes infinite.

108

For sufficiently large samples the quantities

y and a may be replaced by their limiting values",

n n

This gives the somewhat simpler estimators, for

computation purposes

It is shown below that the net effect of this

simplification is to diminish the bias and to greatly

understate the relative efficiency of the orderstatistics estimator to the Gumbel estimator. Since

the asymptotic form (E.2) involves simpler notation

and is occasionally used In practice, it has seemed

desirable to present this case in detail below (Section

1) and also in the main text. The corresponding

results for the original form (E.X) are Indicated in

Section 2 below and tabulated In Table ¥1.

1. Comparison with simplified

Gumbel estimator

From the estimators (E.2) the following

estimator of can be built up, which will be denoted

by i G

i

This has been done, e.g„, in reference 3s pp® 18l

ff. an4_l88, and in reference 18 , p. 10

„
:

'

V

..

,
109

t3

= u> + P’y

p = X + (y

p

- /) ^ S

x

. (E.3)

This is a function of the n sample values x^ , x^, . xR

and it is desired to find its mean and variance,

and thence its bias and efficiency.

The mean is

E(|

g

) = u +/p + (y

p

-/)

^ E(s)p ,

which can be rearranged to give

E(|

g

) = |p+ pf E(s)-l](y

p -f)p . (E.4)

where g p - u + , E(x) = u + and E(s) is the

expected value of the sample standard deviation, s,

when the sample is from the ^reduced*1 extreme=value

distribution exp(=e”^) e Equation (E e i|) shows that

the Gumbel estimator is biased, (unless E (s) = 'k/./E for

all sample sizes, which seems highly unlikely), with bias

A unb la sEd~~e s t 1 mato r analogous to is

l

o = * + (y

p

" f

) s x/e(s) , for, as in equation (E«4)

E (ia

) = u + /p + (yp -() E { s )p/E (s

)

= u + Py

p = l

p

However, this estimator could not be used in an

actual problem since E(s) is not known. Computation

of this quantity was one of the aims of the IBM

computing procedures discussed in the text.
r
no

b(S0

> = e( ^G } " 6p = &T E(s)-1] (yp-/1p . (e.5)

A

The variance of the estimator is

°2( ^g ) = 4

+ h+ 2(yp^)!!7

oov( *’ 3 x>

^ TI 2

where o_ = ^

o (s) is the variance of the sample

standard deviation for samples from the reduced distribution exp(-e”^); and cov(y,s) is the covariance

of the mean and standard deviation in such samples.

A

The efficiency of could be evaluated by

suitable generalization of equation (2+ 0 19

5

to biased

estimators. The variance in the denominator would

be replaced by the MSE (mean square error)." The

numerator would have to be replaced by a complicated

expression which, for unbiased estimators, would

reduce to 0,LB «> Instead of evaluating efficiency for

the biased estimator Ig* therefore, the discussion

will be greatly simplified by limiting it to relative

efficiency. The relative efficiency of one estimator

* For discussion of mean square error see equation

O4..I 3 ) and accompanying text.
' -

jj ! ' - - . .
Ill

(T^) to another (T2

) Is defined as the ratio of

mean square errors,

MSE ( t

2

) H (T

x ,T2

) = (E.7)

MSE (T

x

)

This ratio has been used as an index of comparison

of two estimators (e.g. reference 7)0 Thus, the

relative efficiency of the order-s tatis tics estimator

A

Cp to the Gumbel estimator £ is, by (4«13) and the

fact that the former estimator is unbiased,

a a MSE (|Q

) a2(| ) + (bias(L ))

2 = MSE (f ) o27

(ftp) J~

(E.6) + (E.5) 2 = (lA)Qm

. (E-8)

where k is the number of subgroups of size m into

which the sample of n is par ti tioned (equation

(^., 23), assuming there is no remainder subgroup), and

the expressions needed for the numerator are given

by the equation numbers indicated.

The key quantities needed in the calculation

of relative efficiencies are, from equations (E.5)

and (E.6), E(s), o (s), and cov(y,s). For general

Thus, n = 10 = 2 x 5 gives k = 2, m - 5l n = 20 = 4x5 gives k = 4> m = 5> n = 30=5 x 6 gives k = 5#

m = 6.
X
112

sample size n, their exact values are given by multiple integrals whose evaluation would apparently

require a prohibitive amount of labor „ Instead , the

following method of empirical sampling was used with

the aid of IBM calculating and tabulating equipment.

The universe of (reduced) extreme values |)(y) = exp(-e was approximated by constructing a population of 12,000 suitable random numbers and punching each number on an IBM punch card. These were

then mechanically separated into 1200 random samples

of size n = 10 and for each sample the mean y, standard

deviations, and their product ys were obtained.

This was equivalent to having a

M populationw of 1200

means, one of 1200 standard deviations, and one of

1200 products of the mean and standard deviation.

It was then assumed that the arithmetic mean of each

of the three populations would be a close approximation

to the mathematical expectations (averages) of the

desired quantities, so that these approximations

could be taken as estimates of the moments 'E(s) s E(ys), From these values, and the relation

y-, / 2 \ _ n-1 2

E(s > = — y

n-1 it ~rT T
. : : .nr . V-v . ' v , ;

‘

- V; n.' ‘i;

'

.

~ r-. t:

; : ,••• ..

y-r •• ;• ; . . -f

<

1

. - . . ->

;

' r ...... . :> > 0 . V: r ...

'.

. .

•
113

the variance

2

a (s) = EC's

2

) - [E (s ) ] “ = “ fE ( s

)

was computed* and also the covariance

cov(y„s} = E(ys) - E(y)E(s) = E ( ys ) - Ye(s),

2 p

The five quantities E ( y) , o (y) , E(s), c^(s)* and

cov (y,s) are shown in Table IX * together with the

corresponding theoretical values that can be readily

calculated

.

In actual use this procedure was modified somewhat* since only one value of each of the desired

quantities would be produced by the 12,000 cards and

1200 samples. This single value would be subject

to the fluctuations of random sampling and would be

difficult to rely on in making inferences 0 This

difficulty was1 met by breaking the n popul at ion®* of

1200 samples into 12 sets of 100 samples and obtaining

12 values of each of the desired moments instead of

only one. These 12 values* although each was based

on fewer samples* served to furnish an idea of how

the single value based on 1200 samples was affected

by sampling variation* Such analysis has provided a far firmer basis for judgement of relative efficiency*

H4

The above procedure resulted in moments calculated for samples of size n = 10. In like manner,

600 random samples of size n = 20 were drawn, after

starting afresh by putting all 12,000 cards together,

but this time only 6 instead of 12 sets of 100 samples

were available, resulting in 6 values of the desired

quantities for comparisoa Finally, the 12,000

cards were reprocessed to yield 400 samples of

size n = 30, giving 4 values each based on a set of

100 samples.

The resulting sets of 12 , 6, and 4 values each

were substituted in the appropriate formulas (Eo5)»

(E06)® (E.8) in order to obtain the relative efficiency

of the order-statistics estimator to the (simplified)

Gumbel estimator. These formulas, all of which depend

upon yp, were evaluated at the probability level P = .95*

All these results are summarized In Table ¥ which shows

the values of the bias, mean square error, and relative

efficiency calculated for each set of 100 samples of

sizes 10, 20, and 30, together with the corresponding

N average values obtained from all 1200 samples combined.

115

For ease of comparison, the relative efficiencies

are also charted in Figure 6®

These results constitute the basis of the

statement in the text that at the probability level

P = ,95, for samples of 20 and 30, the proposed method

has greater efficiency than the Gumbel method using

the» simplified estimator, while for samples of 10

the efficiencies are about the same®
' -•
2 0 Comparison with original

Gumbel estimator

116

The estimator corresponding to £ ^ in (E„3)p

built up from the estimators (E„l) is

A

£ 0 ,„ = u + PyF = x + kn

sx

(E.9)

where

k

n

VTP

a

n

d (E.10)

Here

d = A

and

_ %(JE yp

-y

n

°n,P cr-

’ (E.ll)

is the conversion factor for passing from the

multiplier, d, of s x

in (E.3) to k

n

in (E„9) ° It

is apparent from the discussion at the beginning

of this Appendix that for infinitely large n, b. p =1,

so that (E.9) Includes the asymptotic case. For

finite n, however, on < yn < ff''

* Hence, b

n

pi,

being a product of two factors each greater than 1,

may considerably exceed 1, so that the multiplier

b

n p

in (E.10) and (EJ1) becomes appreciably larger

than the multiplier in (E„3)° Thus, for samples of

10, 20, 30 * computation shows that, for P = „95>s

for example. 117

k

10

= 1.397 d

k

2Q = 1.234 d

k

3Q = 1.173 d e (E * 12

)

The bias of H is, i

G,n

Section 1, in view of (E o 10) s

in manner similar to

b (|G>n ) = t-(y

p

-/) - k

n

E( 3 )]p

= Vtf> E(s)bn»P " 1] P (E.13)

Table VI (columns 2 and 3) indicates that the presence

of the factor bn ^p converts the small negative biases

into larger positive ones.

For the variance we have from (E.9)* analogously

to Section 1,

The corresponding expression (E.6) may be written

Comparison of these two expressions shows, since

cov(y,s) was found to be positive, that replacement

of d by the larger value k

n considerably increases

the variance of the Gurribel estimator. Values of the

variance for the original and simplified estimators

are listed in columns II and 5 of Table Vl„
;

'

•

• = .. . fj: 5

- * . k* i

-

.i .

'

' . ,
118

Comparison of these columns indicates that the variance

of the original estimator can become more than half

again as large as the variance of the simplified

estimators, depending on sample size and probability

P„ The effect is most marked for the lower levels

of P and smaller sample sizes and disappears as shown

when both these factors increase,,

The result of the above increases in bias and

variance is to greatly increase the mean square error

(columns 6 and 7, Table VI), and thus to increase the

relative efficiency of the order-s tatistics estimator

(columns 8, 9)« As a result the order-statis tics

estimator is up to twice as efficient as the original

Gumbel estimator even for samples as small as 10

«

This tremendous increase in efficiency falls off

i slowly, as shown, when sample size increases. For

fixed sample size the efficiency increases for

large values of P„ These differences In efficiency

are sufficiently large to completely outweigh any

fluctuations of random sampling attributable to the

empirical sampling method of evaluation usedo

It must be conclude^ therefore, that the

original Gumbel estimator is both more biased and much

less efficient than its simplified form, A '"suit.
T
119

comparison of the order-statist ics estimator with

the original Gumbel estimator gives very conservative

results and greatly understates the actual improvement in efficiency of the proposed method over the

method in present use„
..

O • -ri" " - , „ . " •’ ' ‘ ,

,= •• •
No.

1

23

1*56

78910n

12

13

1U

IS

16

17

18

1920

2122

23

120

ESTIMATION OF EXTREMES

(For instructions see Section 5)

Worksheet 1 - Determination of Estimators

! NACA - Langley — Sample III Computers J. 1. Dates 5/29/52

I. SUBGROUP SIZES AND PROPORTIONALITY FACTORS:

n = 23 = km + m' t = km/n = 0.78261 = 3x6+5 t

2 A = 0.20lil6

k = 3 m = 6 m' = 5

t' = m'/n = 0.21739

t' 2 = O.Oii726

MAIN SUBGROUPS:

Weights a^s b^

i = 1 2

(from Table I)

3 1* 5 6 Check sum

a

± = . 355U5 .2251*9 .16562 .12105 .08352 .01*887 1 \ = -.1*5928 -.03599 .07319 .12673 .11*953 .11*581 - .00001

Cfoservations x. in increasing order from i => 1 to i = TO Subgroup x. No.

1 x

2 x

3

*1* x

5 x

6 Check sum 2a^ Sb^x.^

1 .75 .81 .90 1.08 1.20 1.38 6.12 .89669 .20978

2 .75 .80 .88 .90 1.08 1.20 5.61 .85052 .11*168 3 .98 1.00 1.02 1.15 l.3l 1.1*3 6.89 1.06127 .13870

li5

kSum 2.I18 2.61 2.80 3.13 3.59 1*.01 18.62 2.808I18 .!|9016

T = Saix

iA + (2b1x1A)yp = .93616 + .16339 y

p

IIB. REMAINDER SUBGROUP:

Weights a^, b^ (from Table I)

1 2 3 1* 5 6 Check sum

.1*1893 .21*628 .16761 .10882 .05835 .99999 -.50313 .00651* ,1301*5 .18166 .181*1*8 0

Observations in increasing order from i =1 to i =m' x

2 x

3 x

l* „! „l x

5 x

6 Check sum Za’x

1 “ix

i aJ*i

.75 .93 1,01 1.15 1.16 5.00 .90535 .1831*0

T' = 2a^ + (7bjx[)y

p = .90535 + .l83l|0 yp

III. ESTIMATORS:

6 =t f + t'T' = .9291*6 + .1677U yp u = ,929l|6 , p = ,l677ll

121

ESTIMATION OF EXTREMES

Worksheet 2 - Predicted values, confidence band, efficiency, plotting positions

p y

P

PREDICTED VALUES

Ir = .929h6+.l677iiy

p

Cm* = «5 cj CJ <3" $

I

II

68?-C0NFIDENCE

BAND HALF-WIDTH = VVar(£p)

Q =Q /n

LB °

(Q0 from Table III)

EFFICIENCY

(from T able m) EVar(fp)

(1) (2) (3) (U) (5) (6) (7) (8) (9)

t

2/k = .2Ql*l6 t'

2=.0l*726 p = .16771+

Estimate

.36788

of u:0 .9291*6 g .19117 p

2 .2311*0 p2 .01*997 p2 .0375 .01*820 p2 .965

.50 0.36651 .99091+ g .23189 p2 .27870 p2 .06051 p2

.01*13 .05991* p

2 .991

.90 2.25037 1.3069lt g 1.00065 p

2 1.22831 p2

.26231* p

2 .0859 .23235 p2 .886

.95 2.97020 l.lt2768 g 1.51*171 1.9031*9 p

2

.1*01*71 p

2 .1067 .31*777 P

2 .859

.99 It. 60016 1.70109 g 3.27230 p

2

1*. 07062 p

2

.8601*5 p

2 .1556 .71035 p2 .826

.999 6.90726 2.C8808 g 6.9201*1* p

2 8.65173 p

2 1.82176 p2

.2261* 1.1*6361* p

2 .803

Estimate

1 of p: — 0.13196y

2

p 2

|

0.l6665y2

p2 0.03l*82y

2

p

2 — 0.0261*3y2

p

2 .759

PLOTTING POSITIONS

Observed extremes in increasing rank from 1 to n = 23

Rank Observed Plotting Position Rank Observed Plotting Position Rank Observed Plotting Position

r Extreme r/(n+l) r Extreme r/(n+l) r Extreme r/(R+l)

1 .75 .01*17 n 1.00 .1*583 21 1.31 .8750

2 .75 .0833 12 i.ea .5000 22 1.38 .9167

3 .75 .1250 13 1.02 .51*17 23 1.1*3 .9583

i* .80 .1667 li* 1.08 .5833 Sum 23.62

5 .81 .2063 15 1.08 .6250

6 .88 .2500 16 1.15 .6667

7 .90 .2917 17 1.1S .7083

8 .90 .3333 18 1.16 .7500

9 .93 .3750 19 1.20 .7917

10 .98 .1*167 20 1.20 .8333
- r
TABLE 1 122‘

Weights for mnimum-variances unbiased, linear order-statistics estimator

^ n

of percentage points £ p = u + (3 y 5 and variance

P i=X 1 1 P x - P P

^ Var(Cp) ” - (A^rj + B^.

p

+ C

n)£ , for sample size n = 2 to 6S Xj4 x € •••4 X

q

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f§ ©

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0

g

Limiting

efficiency

as P

approaches

1©

These

are

also

the

efficiencies

for

the

estimator

of

the

parameter

p

0

TABLE IV 126

Efficiency of ordersstatistics estimators for various sample sizes

n = km+ra' partitioned into subgroups as indicated for P = *99 s P=1

(Discussed in Section k*b)

n km + m'

EFFICIENCY

n km + mfl EFFICIENCY

P = o 99 P - 1 P = *99 P = 1

(percent

)

(percent

)

2 or k°2 5^o o £*2*7 21 3x6 + 3 8O08 73-6

3 or k°3 68*7 58-8 22 3x 6 + 1* 8l®8 74-9

h or k®ii 75.9 67-5 23 3x 6 + 5 82o6 75.9

5 or k»5 80.3 73o0 24 bxS 83®2 76-8

6 or k«6 83® 2 76o8 25 5x5 80®3 73oO

7f 1x5 + 2 70-5 60o 7 26 l*x 6 + 2 79 o9 72-3

8 v" 1 x 6 + 2 73o3 63-8 27 kx 6 + 3 81o3 74.3

9 1 x 6 + 3 77®7 69 ol 28 1* x 6 + U 81 o9 75.3

10 2x5 80o3 73oO 29 kx 6 + 5 82o? 76 ol

30 5x6 83®2 76o8

11 1 x 6 + 5 8lo9 75oO 31 5x5 + 6 80 e 8 73o7

12 2x 6 83o2 7608 32 5'x 6 + 2 80o5 73*1

13

c 2x5 + 3 77®3 69 el 33 5x 6 + 3 8l ®6 74-7

Ilf. S 2x6 + 2 7U.7 66 e? 3U . 5x 6 + 4 82o3 75-6

15 3x5 80-3 ?3e0 354 7x5 G0o3 73«0

16 2x6 + U 81® 3 7U-2 36 6x6 83®2 7608

17 2x6 + 5 82o3 75-6 37 7x5 + 2 78 o 2 70-3

18 3x6 83® 2 7608 38 6x6 + 2 80o9 73-7

19 3x5 + h 79®3 71*7 39 6x6 + 3 81 o9 75oO

20 kx 5 80®3 ?3«0 liO^ 8x5 80®3 73o0

61 11x5 + 6 8O06 73®3

If partition is 7 = lx h + 3S then efficiencies ares P = o 99, effo = 72® 7^1 P = 1, eff„ = 63*k%

v If partition is 8 = 2xk$ then efficiencies ares P = * 99 , eff® = ?5®9$| P = 1, effo = 67 * 5% ^ If partition is 14 = 2x 5 + then efficiencies ares P = .99, effo = 79d%$ P = 1, effo = 71 *3%

v If partition is 35 - 5x 6 4- 5.» then efficiencies ares P = -99* effo = 82o8#s P = 1, effo = 16*2%

If partition is UO = 6 x 6 + then efficiencies are s P = o 99, effo = 82 M°s P = 1, effo = 75-7$

127 >w$<

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TABLE VII - PART A

PROBABILITIES OF EXTREMES

(Discussed in Section 6.2)

129

Extremes Frequency PMeon ond Standord Deviaiion

x p x

2 • p

Cumulative Frequency m

Plotting Positions m/(N+l)

75 3 — to —

0 0.OS33 t0

‘ 75 .? O. /250 „ Q

• $Q V O . /<£ (a 7 t0 Si 5

• *S £, 0.2SCO f0 9o 7 0. 29 1C,

t0 fo * 0.3333 tc 93 9 0.37SO f0 9% ~ /o 0. 4/6,7 t0 / . oo II 0.4orS3,.

1.51 /z 0.S0OO t0 /02 13 o.s4/c> t0 /.of /4 0.5*33 t0 /.of /s 0.6,250 tr I./5- 1L 0.6,60 7 t0 /./S' / 7 0.7CS3 t0 1 » /

/S 0.750 0 t0 /. 20 19 0.79/7 t0 /. 20 10 0.S333 t0 /.3! 2/1 o.fiso t0 t.3f 22 <?. <7/4 7 t0 1.43 23 0.95*3 t0

—to to Sums: N = 3, 3

Computer: 7M/P.T/

336 3. X(xp)

Arbitrary Mean:

Date Data: NACA - Zfl^/gy g M.

For

Evaluating

the

Probability

of

Extreme

Values

Environmental

Protection

Section

,

Research

Q

Development

Branch

t

by

the

Method

Developed

by

Dr.

E.J

.

Gum

be!

Military

Plonning

Division

,

Office

of

The

Quartermaster

General.

C - 708 1 (2)

I Jan. 1961

130

TABLE VII - PART B

PROBABILITIES OF EXTREMES

(Discussed in Section 6.2)

I. Mean and Standard Deviation:

N = H ^ I ( x p ) = <2.3

<g Z

VTT = 4- 7?Sa 3 Mean: x' = Ll 03 70

Arbitrary Mean: x

0 = Q.

True Mean 7 = LJ2JL2JL n/(N- 1) =

n. Parameters!

cr„ -- J.at/i

I (x*p) «. 2S. 13 2 a U0937 — (x'r = =

<s x

)’ = —

Standard Deviation: s„ = _

/' 05-41

o dSXo

a • I9&

Q.S3S3

l/a=s,/a„ = 0 • ItO 4-

l/(aV

rN)=(l/a)/v

rN = Q . 0376,

TTT Line of Expected Extremes:

x= u! (l/a)y = 0 - 73 ! 7

y: -2.00 0.00

y ( 1 /a ) : ~ • 36,03 0 00

>5709 - u = > 93n

3.00 s4/2

1-4729

: n/a) = 6- a

*V (l/at = o- 93/ 7

- mode

V' NOTE Upper sign lower sign used for maxima,

for minima

o . !%o4

5.00 2.25 4.60 .9030 4059 . X2 9*

1,333 7

NOTE: Volues x, 0 and x l00 are for return periods of 10 and 100 —

Ha f-width of 0.68269 Confidence Band, <rxm cr

j m Vn/

aVTT) = (O-^VN) [(I/o)/-/n|:

4> (x) : .150 .200 .300 .400 .500 .600 .700 .800 .850

O’y.nn VTJ : 1.255 1.243 1.268 1.337 1.443 1.598 1.835 . 2.241 2.585 .0472 -04U 7 ,0477 .0503 .0543 .06,01 .06,90 0*43 .0972

For largest value, A X _ M

= 1.141 (l/a) = o. sort

For next - to - lorgesf value, Ax „.,

= .759 [N /(N- 1)] ( 1 /a) = 0. 136 9

Expected Extreme, in T periods (years, etc.): X T = *10 + (*100 “ *lo) : *100“ * 10 '

T Z T (XK30"X,0 ) X T T Zr Z t(*ioo~*io) *t T Z T ^*IOO~*lo) X r

1 5 1 ft o 60 .781 140 1.144

20 30 6 70 .847 150 l.l 73

25 404 80 .905 200 1.2 96

30 4ft 3 90 .955 300 1 .4 69

35 54 9 100 1.000 400 1.592

40 607 1 10 1.041 500 1.687

45 65ft 120 1.0 78 750 1.8 59

50 .703 130 l.l 1 2 1000 1.990

ce :

a : NAC.A — Langley SamhJe. No. J1L

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TABLE IX

132

Empirical sampling values of first and second moments of sample mean y

and standard deviation s for samples of n - 10s 20s 3-0

compared

with the corresponding theoretical value where obtainable

(Discussed in Appendix E)

Estimate

n - 10

(1200 samples)

n - 20

(600 samples)

n = 30

(1*00 sables)

of Empirical Theoretical Empirical Theoretical Empirical Theoretical

values values values values values values

E(y) 0*5698 0,5772 0*5698 0*5772 0,5698 0*5772

o*C?) 0*1663 0*161*5 0*0881* 0*0822 0*0535 0o 051*8

E(s) 1*1656 1*2211 1*21*59

a

2

(s) 0*1321 0,0775 0,0513

cKy5 §) 0e 0800 0e 01*38 0,0297

133 co

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(x)j f UOTq.OUTLj iJq.TSUQa

(Adapted

from

reference

18,

Figure

z)

135

&4

>» H

•H£>O

Figure

3

.

-

Comparison

of

efficiencies

of

order-statistics

estimator

|

for

pies

of

sizes

2,

3,

4,

5,

6,

or

for

samples

of

any

size

if

broken

into

equal

subgroups

of

2

to

6.

(Data

from

Table

III,

Part

B.J

Developed

from

DrE

J

Cumbers

Environmental

Protection

Section

t

Research

ana

Development

Branch

Extreme

Probability

Paper

,

by

the

Climatology

Unit

Military

Planning

D'vision

f

Office

of

The

Quartermaster

General

136

/-Reduced Vonote IE 25 If tfii IE lifts IE IP' IP'/O.' IB) /m . !£! II IL HIT ',f IP' A, IP IE OS Return Period-T

Figure 4* “ Graphical analysis of a sample of 23 maximum

acceleration increments by method of order statistics

(Data from Worksheet 2)

<NBS>

Developed

from

Dr

E.J

Gumbe/'s

Environmental

Protection

Section,

Reseorch

and

Development

Branch

Extreme

Probability

Paper,

by

the

C

i/matoiogy

Unit

Military

Planning

Division

l

Office

of

The

Quartermaster

General
.

<
138

journal isa xaquino

jox.eoixx.sd pasodojd jo saxocajoj jja jo oj^ea

THE NATIONAL BUREAU OF STANDARDS

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of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

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