NBSIR 74-602

Efficient lillethods of Extreme-Value

Methodology

Julius Lieblein

Technical Analysis Division

Institute for Applied Technology

National Bureau of Standards Washington, D. C. 20234

October 1974

Final Report

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 74-602 EFFICIENT METHODS OF EXTREME-VALUE

METHODOLOGY

Julius Lieblein

Technical Analysis Division

Institute for Applied Technology

National Bureau of Standards Washington, D. C. 20234

October 1974

Final Report

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent. Secretary NATIONAL BUREAU OF STANDARDS. Richard W. Roberts, Director

Contents

Pag

1. Introduction and Purpose 1

2. Best Linear Unbiased Estimator, Sample Size _< 16 2

3. Good Linear Unbiased Estimator, Sample Size Exceeding 16 6

4. Estimators for Very Large Samples 9

5. Examples 11

6. Summary of Instructions for Fitting a Type I Extreme-Value Distribution 14

7. Further Work 16 References , 24

Tables

1. Coefficients of Best Linear Unbiased Estimators (BLUE) for Type I Extreme-Value Distribution 19

la. Efficiency of BLUE Listed in Table 1 21

2. Worksheet for calculating coefficients a^', b.', for good

estimators based on coefficients a^, b^, of BLUE for smaller

sample size, n = 6, m = 4 22

3. Optimum spacings (X.), coefficients, and ratios of asymptotic

efficiencies of u* ind 6* to efficiencies of BLUE for n = 16, for k = 2, 3, or 4 selected observations 23 Addendum Aii

Efficient Methods of Extreme -Value Methodology

Julius Lieblein

This report presents the essentials of modem efficient methods of estimating the two parameters of a Type I extreme- value distribution. These methods are an essential phase of

the analysis of data that follow such a distribution and occur

in the study of high winds, earthquakes, traffic peaks, extreme

shocks and extreme quantities and phenomena generally. Methods

are given that are appropriate to the quantity of data availa- ble—highly efficient methods for smaller samples and nearly as efficient methods for large or very large samples. Necessary

tables are provided. The methods are illustrated by examples and summarized as a ready guide for analysts and for computer

programming. The report outlines further work necessary to cover other aspects of extreme- value analysis, including other distribution types that occur in failure phenomena such as consumer product failure, fatigue failure, etc. Key words: Distribution of largest values; efficient estimators; extreme values; linear unbiased estimators; statistics; Type I distribution.

1. Introduction and Purpose An increasing number of applications involve analysis of what have

come to be known as "extreme values". These follow a statistical distribution that is quite different from that which governs ordinary data considered to come from a normal or Gaussian distribution. Analysis of

1
extreme -value data requires estimation of the parameters of the extreme- value distribution that gives rise to such data.

It is the purpose of this report to present the most improved version

of the essentials of such methodology, and make it available to those carrying-^ut extreme-value analysis or developing computer programs for such purposes. Not all aspects of extreme-value analysis are presented

in this report, only those concerned with estimation of parameters

described in Section 2; neither is much theory given. More detailed

treatment and theoretical development may be found in the sources indicated

herein. Good general surveys that include extensive lists of references

and cover various approaches will be found in [3] and [9].

2. Best Linear Unbiased Estimator, Sample Size <_ 16

a. Type I Extreme-Value Distribution and Its Parameters A set of data x-j^, x^,..., x^ is said to follow a Type I extreme- value distribution" if the set is an independent random sample from a population represented by the cumulative distribution function (c.d.f.)

-(x-u)/b

' Prob {X<_x} = e^ ^-c«<x<°°, - °°<u<°°, o<b<«' (1)

in which u represents the parameter of location and b the parameter of

scale.The designation "extreme -value distribution" is based on the following. If y^, y^,..., y^ are a random sample of data from, say, a normal distribution, then the average value of such a sample also has a normal distribution, whereas the extreme—the smallest or largest—has quite a different distribution, which of course depends on the amount of

2
data, p. However, for increasing p, such distribution approaches one of three types of limiting forms called "asymptotic distribution of extreme values", or simply, "extreme-value distribution". The type

occurring most frequentiv in NBS applications appears to be Type I for largest values, whose c.d.f. is given above, and to which attention will

be limited in this report. (See Addendum for discussion of amount of basic data, p, and sample size, n)

.

b. Estimation of Parameters

Fitting an extreme-value distribution to a given set of data x^, x^,

x^, or what is the same thing, estimating the parameters of the distribution, means performing calculations on these data to obtain a numerical value (estimate) for u and for b. Many different types of

functions, called "estimators", have been utilized for this purpose, with

widely differing statistical properties.

It is generally agreed that an estimator function, f(x-|^,..., x^)

,

should have the following two desirable properties:

(1) Unbiased . On the average, the estimator function, f, which is a random variable with its own distribution, should equal the parameter being estimated, say u, and similarly for a function,

g, used to estimate b, i.e.,

E(f) = u, E(g) = b, "E" denoting mathematical expection. Such estimators are said

to be "unbiased".

(2) Minimum variance . Generally, as more and more data are taken from the extreme-value distribution, the values of the estimator

function tend to be more and more concentrated about a constant

value. If this constant value is the parameter being estimated,
then the estimates thus tend to become more and more accurate with increasing sajirple size. This property of concentration

' or dispersion is represented by the variance of the estimator

function. For a fixed amount of data, n, different estimator

-functions will have different concentrations, and it is evidently

desirable to have an estimator with smallest variance among all unbiased estimator.

Best linear unbiased estimator (BLUE) .* The above two desired

properties of estimators may be incorjoorated in the statement that the "best" estimator of a parameter is one which has minimum variance among

the class of unbiased estimators.

Generally, a lower (non-zero) bound exists, called the "Cramer-Rao

lower bound" (C.R.B.), for the variance of unbiased estimators of a parameter

based on a given sample size, n (see [1]). This lower bound could thus be considered to be a standard against which to compare the variance of any proposed unbiased estimator, f. Their ratio thus cannot exceed unity, and

is generally less. This ratio,

(C.R. BO/ (variance of estimator f) = (Q/n)/V^^^ (f) = Eff(f) (2)

is designated the (Cramer -Rao) "efficiency" of the estimator f for estimating the designated parameter, where Q is independent of n. The concepts of variance and efficiency have important implications

for sample size. The variances of two (unbiased) estimators may be portrayed

as in the following sketch, showing the tendency of the variance to diminish

with increasing sample size, for two estimators of the same parameter.

*Plural is also "BLUE", not "BLUE's".

4
Variance

- Estimator 2 -~ Estimator 1 —^ Sample Size

'^1 ''2 Estimator 2 is the less efficient estimator, having greater variance, at any sample size, than Estimator 1. It is apparent that a given variance V (or a given accuracy) requires a smaller sample size, n^, with

Estimator 1 than with Estimator 2, for which the sample size required is n^ > n^. The corresponding situation for efficiency of the same two estimators is indicated in the next sketch, showing its increasing

tendency with increasing sample size. Estimator 2 is less efficient

Efficiency

-

' Estimator 1 Estimator 2

Sample Size

^1 '^2 Estimator 1 at a given sample size, n, and requires an increase in sample

size from n^ to n^ in order to reach the same efficiency, E. Thus the variance and its related efficiency have a bearing on sample size, and the most economical estimator to use is the one v\rith smallest variance, or, what is the same thing, with greatest efficiency.

In the case of sampling from the extreme-value distribution, an optimally efficient estimator (i.e., E = 1) has not been found for either n or b. However, recent investigations have shown how to obtain estimators

based on "order statistics", which have very high efficiencies.

5
Henceforth, the sample values will be assumed arranged in ascending

order

:

The x's are then called "order statistics". The desired estimators o£ the extreme- value parameters then take the form of a linear function of the order statistics: ; v v ... n . n u..= Sa.x., b.-.= Zb.x. (4) (n)

^^-L

1 1' fn) .^^11

The BLUE are then obtained by finding numerical coefficients, a^, b^, for which each of the two linear functions of the order statistics is an unbiased estimator of u (or b) having minimum variance. These values,

presented in Table 1 for n 16, were obtained from [11]. Their effi- ciencies are given in Table la based on [8] which shows that relatively

little improvement is obtainable by increasing sample size much beyond

10_, at which efficiency, for u, has reached 98"6, and for u is 851.

3. Good Linear Unbiased Estimator, Sample Size Exceeding 16 The list of coefficients for increasing sample size beyond 16 would

involve increasingly cumbersome tables to use. Instead, a method has been developed in [6] that produces the coefficients of an estimator, for any

sample .^.ize, that is considered to be the best obtainable on the basis of

the knovm coefficients of a best estimator (BLUE) for a smaller sainple size. This method is as follows. Let the known coefficients for a sample of m < n observations be denoted by a^ and b^, giving the following BLUE for estimating u and b

6
from a sample o£ size n: m . m

Then coefficients a11^ I and bl for good estimators ul . and bl- . to take

(m) (m) the place of the a's and b's in (4) are given by (see [7]). m

a: = S a • (t/i)p(n,m,i,t) , (6a)

^ t=l ^

i = 1,2,. ..,n m

b! = E b • (t/i)p(n,m,i,t), (6b)

^ t=l ^ where

P(n,TM,i.t) =

(\] (-j)/0

is the hypergeometric probability function tabulated in [5]

.

A small example, for n = 6, m = 4, will illustrate how these calculations are made. A worksheet format is provided in Table 2, which may serve as a guide for computer programming in the case of large samples.

Starting with the BLUE for sajnple size m = 4,

4 ^4

^4) = ^f^Vt' ^4) = ^^^Vt'

with the a^ and b^ shown in the first two columns of the worksheet, the calculations represented in equations (6a) and (6b) are carried out and produce the "good" estimators

6 . 6

(6) ^^-j^ 1 i' (6) i=i 1 1 \ with coefficients (rounded to 5D) as follows, compared to those of the BLUE taken from Table 1 for n = 6:

7
23

4

5--' ^6

0.34067

.24184

.17038 .11902

.08051 .04759

a.

1 (BLUE) 0.35545

.22549

.16562

.12105

.08352 .04887

-0.37241

- .11460

.04190

.12333

.15591 .16586

b.

1 (BLUE) -0.45927

- .03599

.07320

.12672

.14953

.14581 The variances and efficiencies for these estimators are given by

Estimator

u (BLUE)

b (BLUE)

(Variance) /b"

= V

(from [7]) 0.19123

.19117

.13417 .13196

(Cramer-Rao Lower Bound) /b^

(see [6]) 0.18478

.18478

• .10132

.10132

Efficiency

B/V

.96627 .96657 .75516

.76781 According to reference [7], p. 548, the variance of this "expected value estimator" for samples of size n (with regard to either of the parameters) will not exceed (m/n) times the variance of the BLUE for samples of size m, where m is the size of subgroup . It follows that the efficiency will not be less than that of the BLUE for samples of size m, since (see [2]), Eff - Q/n > Q/n

(n) V,^, - (m/n) \^ QAlL= Eff,

V.^_. (m) (n) ^"""^ '(m) '(m) Thus, the efficiency pertaining to subgroup size, m, might be taken as a rough measure of the efficiency associated with this method. Table la, which gives efficiencies for different sample sizes, then shows that little improvement is to be gained by using subgroup sizes greater than

10, a fact already noted above with regard to total sample size, n.
4. Estimators for Very Large Samples The preceding methods become less practical for very large samples,

say 50 or 100 or more. Fortunately, it has been found that it is not necessary to use all the sample values in order to obtain good estimates.

In fact, as few as four suitably selected observations from a large

sample of observations can yield almost as efficient estimates as using

all n . Theoretical development and detailed accounts are presented in

[4]. Some results from this paper are summarized in Table 3. The n sample values are first arranged in order of increasing size

The idea is to find a small number, k, of the sample values ("quantiles")

\' ''n,"--'^, (8)

v/hich "space" the full set of n values in a manner analagous to the way

a median spaces values corresponding to the fraction, A = .50, with half

. of the observations above the median and the same fraction below—the median is then the "50-percent quantile" of the sample. The "optimum

spacing" sought will then yield fractions

o<A^12< < ... <X, <1

k with the n^ in (8) being given by

n. = fnx.} where the brackets denote the next larger integer, e.g., {4.3} = 5, {16} = 17. Xhe (for each k) for producing good estimates given in Table 3 were found in [4] by minimizing expressions for variances simultaneously

9
for estimators of the parameters u and b by means of asymptotic normal

theory results in large samples, under the constraint of unbiasedness

.

This procedure also produced the other quantities in Table 3. The a| and

b| are the coefficients that give the good estimators for u and b based

on the k selected observations, k = 2, 3, 4:

u* = I a*x

:

' -

" i=i

'

' . k

b* = z b*x

. , 1 n. 1=1 1 Asymptotic efficiency is the ratio of asymptotic variance to the CramerRao lower bound. This variance, while not obtained from the exact variances calculated for the indicated order statistics of the sample of n (which

would require very extensive calculation) is based on normal theory which

becomes more and more exact as n increases indefinitely, and gives

surprisingly close results even for reasonable- size samples. Thus,

asymptotic efficiency is a usable measure of goodness to compare with

exact efficiency for those sample sizes where the latter is available. The comparison is made in Table 3 with BLUE for sample size 16,* where

the ratio is obtained to the exact values given in Table la. The relative

efficiency of the optimum- spaced estimators thus obtained for k = 4 selected observations, is more than 90 percent as good for the estimator

u*, and almost 85 percent as good for the estimator b*, as the efficiency of the best linear unbiased estimator obtainable for n = 16. These high

ratios justify calling these starred values good estimators, although not best for the sample size n. What is very useful about them is that, for a given value of k, both the same spacings, K, and the same coefficients, a*^

*The largest size shown in Table 1. -^q
and b^, can be used for any (large) n, and do not have to be recalculated

for each different n. Thus, a very compact table such as Table 3 can

serve for niany cases. As an example of the use of Table 3 in forming

estimators, suppose n = 100. Then, for the k = 4 selected ordered

observations, x , n , x , x , the spacing, using the X., is that

ri^ i given by n^ = f. 03 (100)}= 4, n^ = {.25(100)}= 26, n^ = {.63(100)}= 64, and

n^ = i 90 (100)}= 91 (the cut brackets denoting the next larger integer),

so that the estimators are

u* = .1893 X, + .4566 x_, + .2772 x,, + .0768 x„,

4 26 64 91 b* =-.3127 X, - .1123 x^, + .2607 x., + .1643 x^., 4 26 64 91 These four selected observations thus yield estimators that are almost

as good as the best available from a sample of 16 observations, namely

(from Table 3) 911 as efficient as the BLUE for u, and 84% as efficient

as the BLUE for b.

5. Examples

The first example illustrates calculation of a BLUE using, as data,

artificial random numbers from a known extreme-value distribution.

The second example applies the procedure for large samples to maximum wind

data to illustrate the calculation of good linear unbiased estimators.

Ex. 1

.

The simulated data, representing a sample of n = 8 observations from the extreme-value distribution with u = 4, b = 1, are as follows:

11
i Unordered Ordered, x. Coefficients to Obtain BLUE

^ a. b.

1 1

1 5.41 3.62 0.273535 -0.394187

2 3.70 3.62 .189428 - .075767

5 3.97 3.70 .150200 .011124

4 4.39 3.82 .121174 .058928

5 4.66 3.97 .097142 .087162

6 ~ 3.62 4.39 .075904 .102728

7 3.82 4.66 .056132 .108074

8 3.62 5.41 .036485 .101936 Check Sum 33.19 33.19 1.000000 - .000002 The coefficients a. and b. are taken from Table 1 for n = 8. Before

1 1 calculating the estimates there are a few useful points to be noted. Check sums should be obtained for every column of values. The first two

colijmns should give the same simi because reordering does not change the total. The coefficients a^ should add as closely as possible to unity

and the coefficients b^ should add to zero. It is shown by theory that these are the conditions for the estimator to be unbiased. These checks

are satisfied above. IVhen they are not, one should look for an error in transcription or some other arithmetical or "human" cause. The resulting numerical estimates of the parameters u and b are, respectively,

8 .8 u = Z a.x- = 3.8724, b = E b.x. = 0.4171.

i=l ^ ^ i=l ^

'

If small calculations such as these are done on the modem hand-held or desk calculators that give sums of multipliers simultaneously with the sums of products then the above check sums will be found a very useful protection against error. As a result of this procedure it is estimated

that the sample of 8 values comes from the statistical distribution or model (eq. (1)) -(x-3. 8724)/. 4171 Prob {X ^ x} = e"®

12
From this one can estimate the probability that a given value of x will

not be exceeded, or, conversely, find what x -value will be exceeded a small fraction, such as 1 percent, of the time. Detailed treatment of

such matters, including variability of resulting estimates, involve aspects

beyond the scope of the present report.

Ex. 2 . The second exanple uses a set of maxiinum annual wind speeds from Chattanooga, Tenn. , for the 21-year period 1944 through 1964. This

example illustrates the simplified procedure for "very large" samples given

previously in Section 4. Although the sample size n = 21 may not be considered "very large", it is adequate for illustrative purposes. The 21 unordered values are (miles/hour)

:

53 62 49 59 40 45 50 45 49 63 57 45

53 63 57 42

41 67 52 41

54 Check sum = TW

Placed in ascending order they are the quantities x^, x^,..., x^^; respectively:

40 45 52 57 41 45 53 59 41 49 53 62

42 49 54 63

45 50 57 63

67 Check sum = VWT

Table 3 allows us to make good estimates by using only 2, 3, or 4 selected values out of the 21. Thus, for 2 values, the fractional distances to go in the ordered set of 21 is = .09, = .73, giving n-j^ ={.09(21^ = {1.89}= 2; and = {.73(21)}= {15.3^= 16. Similarly, for k = 4 selected

values, we would have n^ = {.03(21)}= 1; n^ = {.25(21)}= 6; n^ = {.63(21)}= 14; and n. = {.90(21)}= 19. The calculations, based on the values from Table 3,
=41 0.5673 -0.4837 x^^ = 5^ ^ .4327 ' .4857

Ck. sum 98 1.0000 0.0000

I a.*x = 47.9232 b* = Z b.*x = 7.7392

i=l ^ i=l ^ ^

k = 4

^1 -- 40 ^6

== 45 ^4 == 54 ^19 _^

sum 202

a.* b.*

1 1 0.1893 -0.3127

.4566 - .1123

.2772 .2607

.0768 .1643 .9999 0.0000

4 . 4

Z a.*x = 47.9262 b* = Z b.*x = 6.8672 in, - i=l -

'^i

6. Summary of Instructions for Fitting a Type I Extreme-Value Distribution

If we have a set of n maximum values that follow the "double exponential'

distribution, eq. (1), then the procedure for fitting the distribution,

i.e., estimating the parameters u, b, is to first arrange the n values in ascending order

Xj <X2l... ix^.

and then proceed as follows, according to the magnitude of sanple size n:

14
a. n _< 16 Form the product sums n . n

u = T. a.x. , b = Z b-x.

,

i=l ^ ^ i=l ^

^

using the coefficients in Table 1. These give the numerical values of

the best linear unbiased estimators (BLUE) of the parameters u, b. Further details were given in Section 2b.

b. 16 < n < about 50 For this range of n, take subgroup size m = 10 and proceed as follows

(1) Compute the coefficients

n

. a.' = Z [a . (t/i)p(n,m,i,t)]

1 t=l ^

i = 1,2,. .. ,n, n

= Mb. - Ct/i)p(n,m,i,t)]

^ t=l ^

where the a^'s and b^'s (m in number) are the coefficients of

the BLUE for sample size m, given in Table 1, and

rn-i- p(n,m,i,t) = -

is the hypergeometric probability function tabulated in [5]

.

(2) From the above coefficients, compute the estimators n . n

u> V = Z a. 'x. , bl . = E b. 'x.

.

(n) .^^1 i» (n) .^^ 1 1

These give the linear unbiased estimators that are the best available on

the basis of the known BLUE coefficients for samples of size m. Further

details were discussed in Section 3. 15
c. n > about 50 This is the range of large to very large sample sizes. The estimators are obtained by using only a small number, k, of selected values out of the n, and the procedure is as follows:

(1) Take k = 4 and obtain X. and coefficients a.* and b.* from

. 1 XI Table 3, i = 1, 2,. .

. , k.

(2) Compute n^, n^, n^, n^ with n^ = [nX^], the next larger integer

to nX .

.

1 (3) Compute the large- sample estimators

. k . k u* = z a.*x , b* = z b.*x .

. 1 1 n. . , 1 n. 1=1 1 1=1 1 These estimators are unbiased and have remarkably high efficiency

considering that only a very small portion of the observations are used. Further details were discussed in Section 4.

7. Further Work , This report has presented a preliminary brief account of irodem

developments in extreme-value methodology that can yield optimum or good

estimators of the parameters of the frequently occurring Type I distribution

of largest values. Current methodology concerns many additional aspects of extreme- value analysis which would be useful to the applied scientist. The possibilities for further work include the following:

a. Selection of extreme-value model. There are three types of

extreme-value distributions, including the Type I considered

in this report. Modem methodology includes

16
(1) quick graphical aids for selecting the most appropriate

type; there are a variety of such methods that apply in different situations;

(2) conputer methods for handling many sets of data. One study involved analysis of 38 sets of data on maximum wind

speed with sample size varying from 15 to 53. Over half of

the sets were found to follow the Type I distribution. Further work will compare these methods and determine the best ones to be recommended.

b. Fitting the selected distribution. The work of this report can

be extended to include the other two types. They each involve

a third parameter and require more elaborate methods of estimation.

It will be essential to develop computer programming for comparing

the various methods and testing their sensitivity to simplified

approaches. For example, there is a whole class of estimators, called best linear invariant estimators (BLIE) , which are biased, but have efficiencies exceeding those of the optimum unbiased

estimators discussed in this report. Such estimators have found widespread use in reliability studies of many kinds [10]. This

work will involve largest-value data but will be modified to include smallest -value data treatment in order to accommodate

the Type III, or Weibull, distribution of smallest values that is also extremely useful in failure phenomena and reliability studies.

c. Statistical inference from fitted distribution. Specification of

the distribution complete with the fitted parameters implies

considerable information on future events, provided the model

17
continues to hold (and the parameters do not change) . Thus the probability of exceeding any given large value is determinable,

which is of importance in study of high intensity winds, earth- quakes, floods, and natural catastrophes. Other applications

^re extreme phenomena or peaks in general such as traffic peaks,

as well as breaking strength and fatigue failure of materials, consumer product failure, etc. Conversely, values can be determined which will be surpassed with given probability, or

risk, which is of evident value in these fields.

18
Table 1. Coefficients of Best Linear Unbiased Estimators (BLUE) for Type I Extreme -Value Distribution

n i a^

2 1 0.916 373

2 .083 627

7J 1 . OoD

2 .255 lU

3 .087 966

4 1 .510 998

2 .263 943

XJ DoU

4 .071 380

5 1 .418 934

2 .246 282

3 .167 609

4 .108 824

qJ . Udo odu

6 1 .355 450

2 .225 488

3 .165 620

4 .121 054

5 .083 522

6 .048 867

7 1 .309 008

2 .206 260

3 .158 590

4 .123 223

5 .093 747

6 .067 331

7 .041 841

8 1 .273 535

2 .189 428

3 .150 200

4 .121 174

5 .097 142

6 .075 904

7 .056 132

8 .036 485

b, n i

1 — —

-u. /Zi o4o Qy

1i .ill o4o z

7

. OoU o41 A

M- 91; c: oiO CD

77/1 771;

/ Zo AU

7

. DDO A1 Q Q0 HQ . Uoo Qy .223 919

.248 797 11

2

- . dUj 1 77 iZ / 3 AHA Do4 /I4

1 xr\

. iou /I t: c 40D cJ

1 ft!

. iol DoO A0

1 /1Q7 40O 7

0

/I c:q

. 4oy 777

0Qy

. UO J QQ7 yyz 1 u 077 1 QQ lyy

1 7A 77/1

/ Z4 1

1

1

1

11

1 y1 Q . i4y C 7/1 D04

. 145 c n7 ou/ 7

/I4

- .423 700 cJ

- . UOU oyo 6

. uio 1 no iyz 7 nQ7 . Uo / 77Q ooy Q0

11/1

. ii4 000 Qy

1 7 c oby iU

1 7n 1/11 1

1

11

. jy4 1 87 1 7 iZ 1

- .075 767 9

z .011 124 7 .058 928 4 .087 162 5 .102 728 6 .108 074 7 .101 936 89 10

11

12

0.245 539 -0.369 242

1 7/1 .1/4 ooZ 887 08 ZU707J .141 789 - .006 486

.117 357 .037 977

.097 218 .065 574

.079 569 .082 654

.063 400 .091 965

.047 957 .094 369

.032 291 .088 391

.222 867 - .347 830

.162 308 - .091 158

1 77 8/1 c: 04D 01 Q . uiy ZIU 71 0 .112 868 .022 179

.095 636 .048 671

.080 618 .066 064

.066 988 .077 021

.054 193 .082 771

.041 748 .083 552

.028 929 .077 940

.204 123 - .329 210

.151 384 - .094 869

.126 522 - .028 604

. lUO 77A ZZO 01 0 . UlU 077 UOZ .093 234 .035 284

.080 222 .052 464

.068 485 .064 071

.057 578 .071 381

.047 159 .074 977

.036 886 .074 830

.026 180 .069 644

.188 361 - .312 840

.141 833 - .097 086

.119 838 - .035 655

.103 673 .000 534 .090 455 .024 548

.079 018 .041 278

.068 747 .053 053

.059 266 .061 112

.050 303 .066 122

.041 628 .068 357

.032 984 .067 671

.023 894 .062 906

19
Table 1, Coefficients of Best Linear Unbiased Estimators (BLUE) for Type I Extreme -Value Distribution (continued)

i a b. n i a b —

1 — — 1

I u . ± / 't Q1 A yi 0 - . 298 313 15 1 0. 153 184 -0U . Z777/ o AOA ouo

7 . loo /177 - . 098 284 L7 . xxy "^1

4

OXH- . 008 uyo 7A8 / Oo

3 -— , ±io / Dy - . 041 013 3 . 103 196 . 048 U^fo 78c; ^oo

4 flQQ - . 006 997 4 384 01 7 . UX / Q74 yo4

5 . uo / . 015 836 5 . 081 767 . 007 uuz. 777 / / o 5 077 oDo . 032 014 5 . 073 495 01 7 . UX / 770 / / y

7 . UOo 7A4 . 043 710 7 . 066 128 . 078 UZo Q88 yoo

8 . UO-7 . 052 101 3 . 059 401 . 077 UO / toz, 9 047 . 057 862 9 . 053 140 . 047 U'+O 7Q8 / yo 10 r\AA . 061 355 10 . 047 217 0/1

8

. Uho 41 c; HlO

1

1

X -L . U J /

1 77 . 062 11 . 041 529 oi;i

. UoX c;74 oOf

1 7 . uzy 7Q0 / yu OfSl • uUX oyy . UOO Q84 ot;7

. Uoo £0 /

i- J n? 1. UZi yob . 057 oou • .030 484 0Q7 . Uoo A07 DUo

14 . 024 887 . uo^ OOH

1X

1 fit ouy ox u xo . muxo8 8Q4 OyH . 048 U4o A48 O'+O L7 . IZo yoo 77"^ / / o

. 1 Uo 770 - . 045 120 16 1 . 144 271 7A7 QQO yyu

4 . uyo 77"? - . 013 uo J 2 . 113 346 . 0Q8 uyo 40A 'lUO

. UOH- f^^ 01 Qy . 008 690 3 . 098 600 Oi^O . UoU 771 / ol

5 07 c; .11/3 . 024 787 4 . 087 801 071 . UZl 077 yoo

7 0A7 ool . 035 768 ... 5 f

' . 079 021 007 lO1 A7/ 8 . uoy oOO . 044 767 > • 6 . 071 476 01 7 . UIZ Z770/ U 9 oc? oyi . 050 418 ' '

• 7 1 . 064 771 0? 7 . UZo lOo 10 L\}\) . 054 674 > 8 . 058 660 . 071 UOX ^78

1

X

. 039 847 . Uo / UOO q . 052 Q8Q . 037 939

12 .033 526 . 057 829 ' 10 .047 646 ]042 787

13 .027 131 . 056 652 11 : .042 539 .046 308

14 .020 317 .052 642 12' .037 597 .048 646

^ 13 .032 748 .049 860

14 .027 911 .049 912 15 .022 969 .048 602

: 16 .017 653 .045 207

20
Table la. Efficiency of BLUE Listed in Table 1 n Cramer- Rao Efficiency for Estimators of

L 0. 84047 0V/ • 42700 / \J\J 3 .'91732 .58786

4 .94448 .67463

c . 95824 . 72960

AVJ . 96654 _ 75732

7 . 94315 . 79606

8 .97605 !81785

9 .97903 .83519

10 .98135" .84938

11 .98312 .86119

12 .98474 .87121

13 .98600 .87982

14 .98706 .88725

15 .98800 .89381

16 .98880 .89961

21
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23
References

1. Cram6r, Harald, Mathematical Methods of Statistics , Princeton University

Press, Princeton (1946), pp. 479-480.

2. Goldstein, N. , "Random Numbers from the Extreme Value Distribution,"

Publications de I'Institut de 1 'University de Paris , Vol. XII, lascicule 2 (1963) 137-158.

3. Harter, H. Leon, Order Statistics and Their Use in Testing and Estimation , Vol. 2, Aerospace Research Laboratories, Wash., D.C. (1969), 125-131.

4. Hassanein, K. M. , "Estiination of the Parameters of the Extreme Value

Distribution by Use of Two or Three Order Statistics," Biometrika , Vol. 56, 429-436 (1969).

5. Lieberman, G. and Owen, D. , Tables of the Hypergeometrie Probability

Distribution, Stanford University Press, Stanford, California,

'UMTT.

6. Lieblein, J., "A New Method of Analyzing Extreme-Value Data," NACA

Tech. Note 3053, National Advisory Committee for Aeronautics,

1954, Table III (a).

7. McCool, John I., "The Construction of Good Linear Unbiased Estimates

from the Best Linear Estimates for a Smaller Size," Technometries , Vol. 7, 543-552 (1965).

8. Mann, N. R. , "Results on Location and Scale Parameter Estimation with

Application to the Extreme-Value Distribution," Report ARL 67-0023,

Aerospace Research Laboratories, 1967, Table D.II.

9. Mann, N. R. , "Point and Interval Estimation Procedures for the Two- Parameter Weibull and Extreme-Value Distributions," Technometries . Vol. 10, 231-256 (1968).

10. Mann, N. R. , Schafer, R. E. and Singpurwalla, N. D. , Methods for the Statistical Analysis of Reliability and Life Data , John Wiley, New York (1974).

11. White, J. S., "Least -Squares Unbiased Censored Linear Estimation for the Log Weibull (Extreme Value) Distribution," J. Indus t. Math. Soc,

Vol. 14, 21-60 (1964).

24
ADDF,KTOI Consideration of Sample Size

As pointed out in Section 2a above, there may be two sample sizes involved in an "extreme -value" situation, namely, (i) the amount of data,

p, from which an extreme is taken; and fii) the number of such extreme

values, n, each extreme value in a sense representing a different set of

data from the same population. It was also indicated that extreme -value analysis in theory depends upon an as>Tnptotic situation, one where each

amount of data, p, is "large". It would be useful to have some guide

lines as to how much is large.

This has always been recognized to be one of the most difficult

questions by workers in the field, from the pioneering efforts of

Professor E. J. Gumbel [b] to the present day, and the question is still largely open. As an illustration of how p and n may be considered, take Example 2 of the text above, dealing with maximum annual wind speeds. Records of wind speed are obtained by means of continuous -recording instruments

throughout the year. Maximum values for five-minute periods are read and

the largest of these is taken as the single maximum for the year. Thus

the amount of data "in back of" the year's maximum is represented not by

the 365 daily maxima, but by the much larger number of five-minute

periods. Thus, p is at least several thousand, and there is little

argument that this is a large enough amount of data. In the example,

sample size n = 21, and this is the distinction between p and n. What may

be considered additionally is whether the maxima of all the five-minute

periods are from the same population, as this is a basis for the A-1
theoretical derivation of the extreme -value distribution. Here, discussion must be heuristic, in the absence of definitive studies. Long

experience has made it seem likely that considerable departure from such

a "stationarity" assumption can be tolerated without being detrimental to the application of extreme-value methods. It is as though the limiting process implicit in use of the extreme-value distribution operates to "smooth" out irregularities in the fundamental data, which may not even

be accessible, and is not essential to application of the methods. At any rate, it is true that in the cited example, and in many other such cases analyzed by Simiu and Filliben [f] of the National Bureau of Standards, the maxima appear to follow the extreme-value distribution. A large number of other successful applications will be found in Professor Gumbel's definitive book [b] , substantial portions of which

have been updated by Mann, Schafer, and Singpurwalla [d]

.

It was stated above that the basic data behind the maxima may not

even be available, yet this need not impair the application of extreme- value methods. Many successful applications where "basic data" are not available occur in the fields of reliability and in failure phenomena of materials, products, structures, and systems in general.

This can be illustrated by the simplest case, tensile strength of materials. It is epitomized by the saying "A chain is no stronger than

its weakest link", characterized as the "weakest link" hypothesis. The

idea is that a bar of, e.g., steel is made up of many hypothetical small segments, xvlth tensile strengths represented by a probability distribution. When the bar is subjected to tensile stress, its failure stress is determined by the strength of the weakest of its "segments", which acts
as a "weakest link". This idea is widely attributed to Griffith, in his theory of flaws enunciated in 1920 [a] ; and the first statistical treat- ment based on this, to Pierce in 1926 [e] .— A practical problem involving tensile strength is, knowing the strenght of a bar of given length, can we predict the strength of a bar,

say, twice as long? The answer is generally Yes, by considering the larger bar as though it were made up of twice as many "small segments"

as the half-size bar, i.e., the "amount of data, p" is twice as much, even though the value of p remains unknoi\'n and hypothetical.

Thus, many practical problems are successfully tractable without

ioiowing the amount of "basic data, p". This may be a reason that little

attention has been given to this aspect of the methodology.

The methodology presented in this report is primarily intended for workers who are already applying extreme -value methods to their problems

and may be using methods of estimation that may not be the most efficient

and best available in the present state-of-the-art. These workers have

found, through other methods, outside the scope of the present preliminary

report, that the extreme-value distribution appears applicable to their

problems. This would imply, again on a heuristic basis, that the amount of basic data, p, known or hypothetical, is adequate. At present, only the most general guide lines can be given as to how mall a value of p would be adequate. Thus, if it is known that the

— However, the writer of the present report has found evidence to justify

earlier priority, namely, to Chaplin in 1880 (see Lieblein [c]).
basic data (in amount p) come from a simple exponential distribution

(a failure model used in roany reliability studies [d]), then p can be as small as 5 or even less. However, for some other distributions of basic

data, such as the normal (Gaussian), it is known that such small values of p are inadequate. On the other hand, it is felt that generally a value of p of several hundred is probably sufficient, while for much

smaller values the situation is doubtful. But, as already indicated,

this is a matter of conjecture and requires considerable further research.

In any case, if the user of extreme-value methods finds such methods

applicable in a given case, he generally need not be concerned about the value of p.

A-4
References

a. Griffith, A. A., "The phenomena of rupture and flow in solids,"

Phil . Trans . A, Vol. 221, 163 (1920).

b. Gumbel, E. J., Statistics of Extremes , Columbia University Press, New York (195^7:

c. Lieblein, J., "Two early papers on the relation between extreme values

and tensile strength," Biometrika , Vol. 41, Parts 3 and 4, 559-560

(December 1954).

d. Mann, N. R. , Schafer, R. E., and Singpurwella, N. D. , Methods for the Statistical Analysis of Reliability and Life Data, John Wiley, New York (1^74).

e. Peirce, F. T. , "Tensile tests for cotton yams. V. "The weakest

link- -theorems on the strength of long and of composite specimens," Journal Textile Inst. Trans ., Vol. 17, 355 (1926).

f. SiiTiiu, E. and Filliben, J. J., "Statistical Analysis of Extreme Winds"

(Techjiical Note in preparation)

.

USCOMM-NBS-DC
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2. Gov 't Acrcs.sion 3. Recipient s .^cces.sion No.

4. TITLE .AND SUBTITLH Efficient Methods of Extr^e-Volue Methodology

5. Publication Date O^tooeir 1974

6. Performing Organization Code

7. AUTH(JRIS) Julius Lieblein

8. I^t*rtorminn Organ. Report No. N3SIR 74-602

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16. ABSTRACT (A 200-word or less {actual summary ol moat significant information. If document includes a aignificant bibliography ot literature survey, mention it here.) This report presents the essentials of modem efficient methods of estimating the two parameters of a Type I extreme-value distribution. These methods are an essential

phase of the analysis of data that follow such a distribution and occur in the study

of high v/inds, earthquakes, traffic peaks, extreme shocks and extreme quantities

and phenomena generally. Methods are given that are appropriate to the quantity

of data available—highly efficient methods for smaller samples and nearly as efficient methods for large or very large samples. Necessary tables are provided.

The methods are illustrated by examples and summarized as a ready guide for analysts

arid for computer programming. The report outlines further work necessary to cover other aspects of extreme-value analysis, including other distribution types that occur in failure phenomena such as consumer product failure, fatigue failure, etc

17. KEY *ORDS (six to twelve entries; alphabetical order: capitalize only the first tetter of the first key word unless a proper ndme. eparated by .temicolons) Distribution of largest values; efficient estimators; extreme values; linear unbiased

estimators; statistics; Type I distribution.

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