NBSIR 75-637

Note on Simplified Estimators for

Type I Extreme-Value Distribution

Julius Lieblein

Technical Analysis Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

December 1974

Final Report

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 75-637

NOTE ON SIMPLIFIED ESTIMATORS FOR

TYPE I EXTREME-VALUE DISTRIBUTION

Julius Lieblein

Technical Analysis Division

Institute for Applied Technology

National Bureau of Standards

Washington, D. C. 20234

December 1974

Final Report

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent. Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

NOTE ON SIMPLIFIED ESTIMATORS FOR TYPE I EXTREME-VALUE DISTRIBUTION

Julius Lieblein

Methods for extreme-value analysis (for the Type I extreme -value

distribution) that have optimum properties involve up to 20 quantities

(depending on sample size) whose values are known to 6 decimal places.

The present note shows how to modify these to much simpler values involving

2 decimal places that are more convenient to use yet sacrifice very little

of the optimum features.

Key words: Simplified estimators; linear unbiased estimators; bias; efficiency;

extreme values; Type I distribution; statistics.

1. Introduction

An NBSIR by the writer [1] described the occurrence and nature of the

Type I extreme-value distribution and presented estimates of the two parameters

of this distribution for various ranges of sample sizes from very small to very

large. It was explained that for any sample size there exists a BLUE- -best

linear unbiased estimator- -with optimum properties. These estimators are linear

functions of the sample order statistics- -observations arranged in ascending

order. The coefficients of such estimators were given to sample size n = 16,

and are known to n = 20, to six decimal places.

For rapid and convenient use, it seems desirable to try to replace the more

exact six-decimal coefficients by much simpler values, with two or even one

decimal place or significant figure. It is the purpose of this note to show how

to obtain such "simplified estimators" with properties almost as good as the

more exact best ones. For this it will first be necessary to present the

expected value and variance of linear forms, as related to the extreme-value

distribution.

The linear (order statistics) estimators of the parameters u, b are:

u = n

i=l a-x.

1 1 (1)

b = n

i

i=l

b.x.

or

I u

C = I ni = C X. (la)

n
where C is the nx2 matrix of coefficients (prime denotes transpose) , x is

the n- rowed vector of the n observations, after arrangement in ascending order

(order statistics) i.e.,

x^ < X2 5 ... <

Before ordering the x's are independent observations from the Type I extreme value distribution

(x-u)/b

Prob. {X < x} = e

-oo < X < °°

-oo < U < °° 0 < b < °° (2)

Tlie expected values of the estimators (la) are given by:

'

-

^ E(c) = C'E(x)

.

In order to proceed, the random variables Xj^ must be expressed so as to

exhibit the parameters explicitly. This is done by writing each x^ as:

(3)

X

i = u + by^, i = 1, , n. (4)

where the y. are the n order statistics from the "reduced", parameter- free

distribution (corresponding to the standardized distribution in the normal

distribution case)

-y

Prob. {Y < y} = e -oo < y < 00 (5)

Eq. (3) then becomes

E(c) = C (ul + bE(y))

f 1 Ey^

= C

1 Ey

n

= C e c. (6)

2
where 1 is the nxl vector of I's, E(y) is the nxl vector of the known expected

values"of the order statistics, y^; e is the nx2 matrix with column 1, E(y),

and c is the 2x1 column vector of the parameters u,b.

For unbiasedness , the expected values of the estimators (6) must

equal the parameters. The conditions for this from (6) are:

c, or (C'e - I2) c = 0, i.e.

,

(7)

n n

Ei=l

a1 = 1, 0 (7a)

n

Z i=l

b.

1 = 0, 1 (7b)

The BLUE are unbiased, and unique, being "best", by definition and calculation.

Therefore any alteration such as simplified estimators would result in bias.

However, the variance may be less, since we are no longer restricted to the

class of unbiased estimators. The measure of goodness of the estimator must

then be modified to include the bias; it becomes the mean square error

of the estimator about the parameter estimated, not about its expected

value as is the case with the variance, i.e.

MSE(u) = E(u - u)"^ = E[(u - Eu) + (Eu - u)]"^

= E(u - Eu)^ + (Eu - u)^

'

= VARIANCE (G) + [BIAS (G)

] ^ (8)

the middle term on expanding the square vanishing because it is a multiple

(namely, (Eu - u)) of:

/\ /\ /\ ^ E(u - Eu) = Eu - Eu = 0.

For unbiased estimators, MSE and variance are the same.

2. Bias

For biased estimators, the bias is given by, in place of (7a) and (7b),

(C'e - I2) (see (7)), i.e.

^ n n

bias(u) = ( Z a. - l)u + ( Z a-Ey.) b (9a)

i=l ^ i=l ^

^

^ n n

bias(b) = ( Z b.)u + ( Z b.Ey. - 1) b (9b)

i=l ^ i=l ^

^

3
Tlie MSE is thus, in general, a quadratic function of the two unknoMi

parameters u and b and so presents a difficult situation. To make it more

tractable and reach definite results, we make adjustments, which will

usually be small, in the coefficients of the simplified estimator, so that

the parameter u will not appear in the bias.

3. Simplified Estimators

a. Construction

The simplified estimators are summarized in Table 1 for n = 10.

The first column gives the coefficients of the BLUE estimators for u and

for b. Col. (2) gives the BLUE coefficients rounded to 2 decimal places.

The a's add to 0.99 instead of 1.00 as would be necessary in (9a) for the

u-term to disappear, so a slight adjustment is made that would least affect

a coefficient—in this case, a^ is increased by the minute amount, 0.0012,

which permits rounding to .01 more and so raise the total to 1.00 (Col. (3)).

Also, it turns out that the b's add to 0.00 so no adjustment is necessary

there. Another type of rounding is to 2 significant figures instead of 2 places, and this time adjustment is necessary in both an a- and a bcoefficient. Cols. (4) and (5). The next 4 versions. Cols. (6) to (9), are formed similarly on the basis of 1 decimal place and 1 significant figure.

b . Variance, MSE and Efficiency Ratio

Using the "propagation of error" fomula for variance of a linear

form (see [2]),

var(L'x) = L'V(x)L, (10)

where L and x are n-rowed column vectors of coefficients and variables,

respectively, and V(x) is the nxn matrix of variances and covariances of the

x's, we have

var(u) = a'Va = (a'va) b^ (11)

var(b) = b' Vb = (b* vb ) b^ ^ 0 0 ^0 0^

where by (4) , '

V(x) = v(y) b^ ' (12)

with v(y) the nxn variance -covariance matrix of the reduced extreme value

order statistics, y. , and the arguments x and y are suppressed for

convenience; the quantities a and b are the n-rowed vectors of the

coefficients a^ and b. respectively? (The subscript "o" is used to avoid

confusion with the parameter b)

.

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From (11) and relations such as (8), we have:

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n

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i=l ^ (13a)

MSE(b) = b' vb + o o

n

( ^ 1=1

b^Ey.) (13b)

Calculation of bias, variance, MSE were carried out by use of

OMNITAB on the MBS 1108. A copy of the program is attached, and can be

readily modified to give results for any other sample sizes where the

BLUE coefficients are known; at present they are known* for sample sizes

up to n = 20. They are shown in Table 2 for n = 10. The 4 adjusted

estimators (Col. (1)) are those in Table 1, Cols. (3,5^7,9) as indicated.

Bias (Col. (2)) is in terms of b only, as shown, since the term in u has

been suppressed through the adjustment. Variance and mean square error,

in terms of b , are shown in Cols. (3) and (4). Col. (5) gives the

"efficiency ratio", which shows how the "efficiency" measure MSE compares

with that of the "best", BLUE. (A ratio greater than 1 means BLUE is

more efficient, and vice versa .

)

For example, when the estimator is simplified and adjusted to

2 decimal places as described above ("2D") , the efficiency is virtually

the same for the estimator of the parameter u, and only about 1/2% worse

(larger MSE) as shown in the fourth line of Column (5) in Table 2. For

two significant figures (the third estimator) , the results are virtually

the same as for two places. If the estimator is altered still more drastically,

to 1 figure—whether decimal or significant—the efficiency becomes worse,

as might be expected. Similar remarks apply to the amount of bias, being

virtually mil with two-figure estimators, and more appreciable with one- figure estimators.

These results make plausible the following statement, for

sample sizes that are not too small, say 6 or more:

Two-figure coefficients (whether 2 decimal places

or 2 significant figures), for estimators of the

two parameters of the Type I extreme -value

distribution, can yield practically as good

efficiency as is obtainable by BLUE.

*See reference for paper by White [1]

.

6
Table 2. Bias and Efficiency of Simplified Estimators Adjusted

so Bias Depends only on b, not u, n = 10 (upper line

relates to u, lower line to b)

E: :ficiency Ratio

MSE/b2

9

flVlSF /h var(BLUE)/b^

(i) (2) (4) (5)

BLUE 0 0.112973 0.112973 1

0 .071573 .071573 1

Simplified Est.

Adj . to 2 D 0.000458 .113002 .113002 1.000256

(Col. (3)}* .002725 .071973 .071980 1.005686

Adj. to 1 D -.050827 .113345 .115928 1.026157

(Col. (5)) .192893 .102266 .139472 1.948668

Adj. to 2 S -.001240 .112926 .112927 0.999593

(Col. (7)) .003570 .072085 .072097 1.007321

Adj. to 1 S .044711 .117207 .119205 1.055164

(Col. (9)) -.103800 .057531 .068305 .954340

*Column numbers refer to estimators in Table 1.

7
REFERENCES

Lieblein, J. "Efficient Methods of Extreme-Value Methodology,"

NBSIR 74-602, October 1974.

"Generalized Propagation of Error Using a New Approach," Proceedings

11th Annual Meeting, Institute of Nuclear Materials Management , May

25-27, 1970, sec. 2.1, p. 192.

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1

• -0 coy .:- 1. pvni |. A

1 ,,0,1 ols Ki-l'oH r NO.

s ,c-T resiR 75-6:37

2. '..,v'r A.x-c:.

Noir.G on Sia\;-lified Est.Lmators for Type I Extri'.ae-Valuo

Dii3tr.lbutio I

5. Hublu .1, Im., ii.trc

Dece.rh:;- .197/^ . 6 . ! t.' I i ' >i 1 i i i i.' '

' I' t.' t () 1 >' -I I i-< * 1 : ^ * l'

. ' . . ^ . . . .

J 1

I JtLli.vLS.....Lj. eble.ir. . .

S. l-'c;^..; -n^ i Hc;itirr NBSC- 7S-G.^7

9. PhKi-o:; \.iNc; OR{i.~:-;i/..'\Tio.N kasii-: and aodki ss

H.ATION\L BUREAU Or STANDARDS

DEPARTMENT Of COMMERCE

10. Projec: 1 -...•.'/ ..'..rk- L'..": N',,. i3i4:iix,.., _

1 1. Conn .icc 'C'f. tilt Xo.

12. -~^r,i'!is;>r;iii; Or>;::r. vaUvip, Naip.c lull (onipletc Ad.jlrjr (Stre.jt, City, Sf,/f.;, ZIP)

Same as Nos 9

13. I'ypc .vf Kcp;)ri i'.Ta .!

j C - ov crc 1,1 !

14. S'- -,i,in.i i; .'V ,11.; \,- C.o.l .-

iS. .sUIT'l.lvMKM'ARY MJTh.S

:5. A15>TR ACf (Vl i(/vi-worcf or /e-ss lacti'.ctl minimnTy ol n:ost significant infomuUion. l( document inc/uc/f.s ^ si-^nificciiit biblioiS.'.iphy or I itc:rature .survey, mr:ntion it here?.)

Methods for extreme -value analysis (for txie Type I extreme-value

distribution) that have optimum properties involve up to 20 quantities

{depending on sa.uiple size) whose values are known to 6 decimal places.

The pre-,ent note shows how to modify these to much simpler values involving

2 deciuUil plj.:,-^s that are m.ore convenient to use yet sacrifice very little

of the ov^JtimLim features.

17. Ki-Y "OtvDS (six to twelve entries; alphabetical order; capital i z j only the fir-it lettor of tht; fir^jt koy word unless j proj>-jr name; Si;parated by semicolons

)

Simplified estimators; linear unbiased estim.ators ; bias; efficiency;

extreme values; Type I distribution; statistics.

18. A . .[L/wJii.lTY Fx Unlimited

[Zi For Oii' A Di. srribution. Do Not Release to NTIS

; Or.:.-:r From Sut^ of Doc, U.S. Govcinment Printin^^ Office

Viashni-ton, D.C. 20402, SD Cat. CH

Q_' Or..i.. r !.; N.xtional Technical Inforrnritioii Service (NTIS)

Sr^CURlTY CLASS

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14

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