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NATIONAL BUREAU OF STANDARDS REPORT

14 March 1952

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e£fo6 NBS Report

i,,V No„ 1521

PROGRESS REPORT FOR JANUARY-MARCH 1952

on

application of statistical theory of extreme values

i T0 TOST-LOAD PROBLEMS

(NACA Project W4712; NBS Project 1103-21-5106)

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issue Approved for public releaS V

tion, reprinting, or re- production in any £o nirprtor 0f the National Institute ted unless permission in

writing is obtained DlfGC

TorhnolOKV (NIST) a l Bureau of Standards,

Washington 25, d. c. standards and Technology on October 9, 2015
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APPLICATION OF STATISTICAL THEORY OP EXTREME VALUES

TO GUST-LOAD PROBLEMS

(KACA Project V/I;712; NBS Project 1103»21-5106)

I. SUMMARY

Work during the present quarter ( January-March 1952) has

befen marked by a discovery that radically alters the basic

Computation techniques for the above project® The new method

involves a simple means of expressing the covariances of the

order ^statistics in a random sample of n observations from the

extreme-value distribution in terms of tabulated functions®

As a result, the scope of the results to be expected can be

greatly extended without much* if any* increase in cost over

what had been planned.

The present report gives a brief summary of what has been

accomplished and what can be expected under the new techniques,

and includes tentative plans for preparing the results for

publication, This report presents only the main lines of development and avoids most of the mathematical details, since

it is intended to serve as a basis for discussion of further

work by representatives of the National Bureau of Standards and

the National Advisory Committee for Aeronautics at a meeting to

be arranged in the near future® The balance of work under this

project, as outlined herein, is to be planned with the goal of

completion by the end of FY 1952®

II o DESCRIPTION OF RECENT WORK

Ao Background®

The specific objective is to obtain a statistical

function which will provide improved methods of estimating

maximum values of acceleration increments and gust velocities which may be expected by an airplane in flight®
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14 March 1952

B 2 ®

By maximum acceleration increment is meant the

largest value occurring during a single flight of a given

airplane o If a series of n flights of the seme plane are

considered, thon there will be a maximum value X for each

and the set of the n maxima, X^, Xg, 0 «o, Xn , constitutes

a sample of n observations to be analyzed c One objective

of the analysis is to predict a value such that on© may,

in a long series of flights, expect that the proportion P

of the flight maxima will not exceed this value, while

the remaining proportion, 1 ^ P, will exceed (or equal)

it 0 This upper limit is designated as Xp and naturally

depends upon P© For example, if we want to estimate a

limit Xp such that in only a very small proportion (1 « P)

of the flights will a larger value occur, then Xp must b©

expected to be quite large© The limit Xp is not known

but must be estimated from sample data such as the set of

n maxima Xi, Xz» Xn mentioned above©

The method of estimation is to find a function

f ~ f(X^, X2 » •• e* Xn) of these n variables, called

an estimator of Xp, which conforms to the following two

desirable characteristics as closely as possibles

(1) The estimator is unbiased ; E[f(Xl, X2, o©o s Xh)

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= Xp, where E denotes mathematical expectation * This

means that the estimator f fluctuates about a longrun average which is the correct value, Xp©

(2) The estimator is most efficient , that is, has

minimum variance: 02(f) = E(f-Xp)^ , where o2 denotes

variance© This means that the values taken by the

estimator f are concentrated so closely about the

desired true value Xp that of all unbiased estimators

of Xp it has minimum mean squared error©

It Is clear that the values of a function f possessing

the above two properties may be expected to give very

satisfactory estimates of the unknown value Xp © However,

In order to apply these two properties, some assumption

must be made about the form of the statistical population

from which the observations are assumed to come© The fact

that each observation is itself a maximum of many individual

values encountered In an individual flight, together with

other supporting data 'discussed by Press (reference 1),

gives theoretical ground for assuming the underlying population to be of the extreme-value type studied by Dr© E© J®

Gumbel, namely F(x) * exp (»e“y ). y

3 a(x-u) , where F(x)

denotes the cumulative distribution function.
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5106 PROG RPT JAN-MAR 52

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Methods of estimation in present use by the NACA

involve the mean and standard deviation of a sample of

given data* These methods have been developed principally

by Dr® Gumbo1 ® They are not very complicated* but have

th© disadvantage that their bias and efficiency have not

been evaluated because of the great amount of calculation

that would be necessary in order to determine th© expected

values and variances of the functions involved in Dr®

Gurobel ? s estimators®

B Current

Research to date under the present project has concentrated on building up a simple type of estimator f

which is a linear function of the order statistics of the

sample® That is* if we arrange the n observations in increasing order of size and let x^ denote the smallest and

Xn the largest, then the sample may be represented by

( X\ , , & o e> , Xj^ ) , — X2 ®c-«» ^ Xjj , where the xi are called order statistics of the sample,

and the linear estimator sought is of the form

Tn

85 wlxl W2X2 • « • + wnxn ° The weights w^ are to be determined so as to satisfy conditions (1) and (2) above, namely:

n

(1) Tn is unbiased: 2 wjExi = Xp ;

^ n n

(2) Tn has minimum variance: o2 (T ) * 2 2 oijwiwj , j~l i«l

is to be a minimum, where the coefficients denote

the variances and covariances of the order statistics

of the sample of n.

The means, Ex^, in the foregoing are the first moments

of ranked extremes and have already been tabulated in

reference 2® It had been planned to compute the variances

and covariances by numerical integration for sample sizes

up to n * 10, above which point the computation would have

become too costly®

The discovery during the present quarter of a method

for expressing the ojj explicitly in terms of tabulated

functions considerably reduces the amount of computation

necessary for small samples®
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— II «•

Larger samples can be handled by breaking them into

smaller samples, obtaining an estimate of the desired

quantity from each one of these samples, and pooling the

results® In this manner samples of any size can be handled

by the techniques developed for very small samples e This

method of subgroups also has the advantage of making

possible a control chart procedure whereby internal con~

sistsncy of the data and stability of operating conditions

could be checked* The method is also especially well

adapted to the form of estimator being investigated, and

would not be applicable to more complicated functions,

such as those of Gumbel.

The new method has been rendered still more powerful

by a refinement devised by Mr. I. Richard Savage of the

Statistical Engineering Laboratory* This refinement not

only makes it possible to determine the unbiased estimators

for all probability levels p simultaneously, rather than

for just a few selected levels such as P - *95, *99 * but

with very little additional effort yields also the estimators of the two parameters which make it possible to fit

an extreme-value distribution to a given set of extreme

data. The method also automatically furnishes the efficiency associated with the estimate obtained.

The method has been tried out on a preliminary basis

with encouraging results. Unbiased estimators have been

found for samples of any size based on splitting the

sample into subgroups of n = 2 and 3. If subgroups of 3

are used Instead of subgroups of 2, the improvement in

efficiency is represented by an increase of 1$ percentage

points, on the basis of ICO percentage efficiency of the

theoretically most efficient estimator. If we proceed from

n “ 3 to n * If, etc., further jumps In efficiency are

expected to take place, with the result that by the time

n = 6 or 7 we should reach 80 or 90 percent efficiency,

which is probably adequate for the purpose in view.

Co Proposed further work.

V/ork thus far points to immediate continuation along

the following lines:

(1) Computation of "subgroup” estimators described

above for n *^5# 6, 7* say, in order to reach a

' practical lev&l of efficiency 0

(2) Evaluation of the bias and variance of the present

Gumbel-type estimators for comparison with the proposed

estimators at least for samples of moderate size. It

is proposed that this be accomplished by methods of

empirical sampling for n ® 10, 20 9 30 o
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(3) Comparison of the results in (1) and (2) w5.th

asymptotic theory*

Asymptotic theory for large samples has been

discussed in NBS Report 1129# prepared under the

NACA project (reference 3) < It was there shown

how to select as few as three out of a large

number n of sample values, which yield unbiased

estimators of surprisingly good efficiency 0 If it

turns out that asymptotic theory compares favorably

with exact methods even for n as low as 10, then

an improvement in procedures might become possible

for larger values of n c IXX 0 PLANS FOR PUBLICATION

The following means of writing up the various aspects of

the work are envisioned, subject to the usual NACA clearance

procedures:

(1) A mathematical paper presenting the derivation of the

formula for the evaluation of the double integrals

mentioned above, together with related mathematical

results o

(2) A technical report to NACA, somewhat in the nature of

a manual, which would express the new method in

practical terms and would be written for the use of

engineering personnel and field workers 0 This report

might also give a general indication of the nature and

characteristics of the method 0

Julius Lieblein

Statistical Engineering

Laboratory

National Bureau of Standards

Washington, D* Co

14 March 1952
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REFERENCES,

1, Press, Harry: The Application of the Statistical Theory

of Extreme Values to Gust-Load Problems,

NACA TN 1926, 1949c

2, National Bureau of Standards: Table of the first momant

of ranked extremes, NBS Report H67 , September

20, 1951-

3, Liable in, Julius: On certain estimators based on large

samples of extremes 0 NBS Report 1129 » August

15, 1951.
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