NATIONAL BUREAU OF STANDARDS REPORT

1129

ON CERTAIN ESTIMATORS

BASED ON LARGE SAMPLES OF EXTREMES

By

Julius Lieblein

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

E. U. Condon, Director

THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of the divisions and sections engaged in technical work. In general, each section is en- gaged in specialized research, development, and engineering in the field indicated by its

title. A brief description of the activities, and of the resultant reports and publications,

appears on the inside of the back cover of this report.

1. ELECTRICITY. Resistance Measurements. Inductance and Capacitance. Electrical Instruments. Magnetic Measurements. Electrochemistry.

2. OPTICS AND METROLOGY. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Gage.

3. HEAT AND POWER. Temperature Measurements. Thermodynamics. Cryogenics. Engines and

Lubrication. Engine Fuels. Combustion.

R. ATOMIC AND RADIATION PHYSICS. Spectroscopy. Radiometry. Mass Spectrometry, physical Electronics. Electron Physics. Atomic Physics. Neutron Measurements. Nuclear Physics. Radioactivity. X-Rays. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Instruments Branch.

5. CHEMISTRY. Organic Coatings. Surrace Chemistry. Organic Chemistry. Analytical Chemistry. Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry. Spectrochemistry. Pure Substances.

6. MECHANICS. Sound. Mechanical Instruments. Aerodynamics. Engineering Mechanics. Hydraulics. Mass. Capacity, Density, and Fluid Meters.

7. ORGANIC AND FIBROUS MATERIALS. Rubber. Textiles. Paper. Leather. Testing and

Specifications. Organic Plastics. Dental Research.

8. METALLURGY. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy . Corrosion.

9. MINERAL PRODUCTS. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Building Stone. Concreting Materials. Constitution and Microstructure. Chemistry of Mineral

Products

.

10. BUILDING TECHNOLOGY. Structural Engineering. Fire Protection. Heating and Air Conditioning. Exterior and Interior Coverings. Codes and Specifications.

11. APPLIED MATHEMATICS. Numerical Analysis. Computation. Statistical Engineering. Ma- chine Development.

12. E LECTRON ! CS „ Engineering Electronics. Electron Tubes. Electronic Computers.

13. ORDNANCE DEVELOPMENT. Mechanical Research and Development. Electromechanical

Fuzes. Technical Services. Missile Fuzing Research. Missile Fuzing Development. Projectile Fuzes. Ordnance Components. Ordnance Tests. Ordnance Research.

IH. RADIO PROPAGATION. Upper Atmosphere Research. Ionospheric Research. Regular Propagation Services. Frequency Utilization Research. Tropospheric Propagation Research. High

Frequency Standards. Microwave Standards.

15. MISSILE DEVELOPMENT. Missile Engineering. Missile Dynamics. Missile Intelligence.

Missile Instrumentation. Technical Services.
NBS PROJECT

August 15, 1951 NBS REPORT

1103-21-5106 1129

ON CERTAIN ESTIMATORS

BASED ON LARGE SAMPLES OP EXTREMES

By

Julius Lieblein

Research supported by the

National Advisory C oirmiittee

for Aeronautics

This report is issued fc in any form, either in wb

from the Office of the Di

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

i, reprinting, or reproduction

lission in writing is obtained

ihington 25, D. C.
FOREWORD

The methods of analysis of extr erne-value data developed

in the present report will he useful in the economical

handling of large amounts of extreme gust-load data for

airplanes In flight 0 These methods will in many cases give

better results than procedures used previously which required

a much larger amount of calculation

The development of these newer methods is one phase of

a project aimed at the improved application of the theory of

extreme values to the analysis of gust loads of airplanes*

This research, carried out by Mr* Julius Lieblein under the

general supervision of Dr* Churchill Eisenhart, Chief of the

Statistical Engineering Laboratory, was supported by the

national Advisory Committee for Aeronautics* 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. H« Curtiss

Chief, National Applied

Mathematics Laboratories

E. TJ. Condon

Director

National Bureau of Standards
ON CERTAIN ESTIMATORS BASED ON LARGE SAMPLES OF EXTREMES

Contents

I. Summary

II * Introduction

III* Symbols

IV. Extreme-value parameters

V« Estimation of mode -u when (3 is known

1 ,

11 Opt imum" spa c ing

a. Use of -one sample value

b* Use of two or three- sample values

2* Equidistant spacing

3 « ' Method * of . expected values

VI® Estimation of dispersion parameter (3 1* Use of two sample values

2* Use of four sample values

VTI* Estimation of mode when both parameters are -unknown

VIII* Conclusions

References

Tables

I* Comparison of three types of methods of estimating

the mode u of the distribution of largest values

II* Distribution of n = 485 maximum gust velocities

lilt Estimates of mode u obtained by various methods

IV* Calculations for estimating mode u from sample of

n = 485 extremes

V. Calculation of spacing constants for expectedvalues method

Figures

1. Density function for extreme-value distribution

with parameters * = 1, u = 0

2 e Specimen of extreme-value probability graph paper
/

<

.

, -

*'

c
ON CERTAIN ESTIMATORS BASED OH LARGE SAMPLES OF EXTREMES

Julius Lieblein*

I, SUMMARY

Statistical techniques are developed and applied for

the economical handling of large masses of extreme gustload data, These methods are especially well adapted for

punch-card equipment, requiring essentially just a counting

sorter. The efficiencies of the proposed methods when used

with large samples are in some cases superior to the efficiencies of certain methods proposed by E, J, Gumbel which

have been used in an NACA publication (reference 1)*

This report is limited to methods applicable to large

samples. The principal technique is the use of simple sums

and differences of order statistics. If the n values in a

sample are arranged in (say) ascending order, they are

called order statistics . If the data may be assumed to come

from a distribution of extreme values, these order statistics can be used in estimating the parameters of the fitted

distribution. This report describes and gives the efficiencies of rapid methods:

(i) for estimating the population mode when the dispersion is known by using 1, 2, 3, 4, or 5

suitably chosen order statistics from the sample

of n;

(ii) for estimating the population standard deviation

by using 2 or 4 suitable order statistics*

These methods are summarized in Table I; their use is explained in the text of this report, and listed in condensed

form under Conclusions e

* Statistical Engineering Laboratory, National Bureau of

Standards. Preparation of this paper was sponsored by

the National Advisory Committee for Aeronautics.
2

II o INTRODUCTION

The purpose of this report is to contribute some results

on the use of order statistics in the analysis of large

bodies of extreme gust~load data® Reference 1 indicates

that the distribution of extreme values of the fora

F(x) 33 exp [ -e"^ x“u )]

35 Probability that an observed value is <

is applicable to problems of predicting the frequency of encountering very severe gust loads and gust velocities under

certain test and operating conditions * Applicability of the

extreme-value distribution will therefor© be assumed for

purposes of this report®

The problem is then to fit an extreme-value distribution

to a sample of data in order to provide a basis for analysis

and prediction® A method of fitting by estimating the two

parameters, u (the mode) and 1/a (the scale factor or dispersion), has been proposed by E® I® Gumbel (reference 2, lecture

3, pc 18) « As a result of preliminary analysis it appears

that this method may have low efficiency relative to the best

efficiency possible, efficiency being measured relative to

the number of observations necessary to assure a specified

degree of precision,.

It is therefore desirable to investigate alternative

methods of estimation for possible improvement in efficiency®
3

To -avoid ambiguity, it is essential to distinguish

clearly between an estimator and an estimate 0 An estimator

of a parameter is a mathematical function of sample values

used to estimate the parameter. An estimate is the numerical value of this function obtained by substituting into it

the values in an actual sample of observations.

One type of approach has been given by F. Mosteller

(reference 3) for large samples from a normal population.

If the n sample values are arranged in (say) ascending orders

X1 * x2 9 0 * • * *n * they ar8 called order statistics with ranks 1, 2, ...n, respectively. Mosteller has found a method of choosing in

advance k of the n ranks, k * 1, 2, . .., 10, such that the

correspondingly spaced order statistics serve to build an

estimator for the mean of a normal distribution which is up

to 95 percent efficient, and an estimator for the standard

deviation which is up to T5 percent efficient.

The methods used by Mosteller have been adapted and extended in this report to give results for extreme ~value

samples analogous to results he obtained for samples from a

normal distribution.

The big advantage in the use of order statistics for

large samples is that three or fever of the n order statistics can give as much precision as the use of all m values

treated by ordinary methods. For example, for n * 1000, the

- •» 200th value in order of size estimates the mode of the
4

population with greater precision than th© average of all

1000 values® For large samples the appropriate ©ample

values can readily be selected from the data by mechanical

sorting, provided the data are available in complete detail,,

before grouping or other processing®

IXXo SYMBOLS

x random variable

F(x) cumulative probability function

t{x) density function,, derivative of F(x)

a scale parameter ©f distribution of largest values

P same as 1fa

u mode of distribution of largest values

n„ N number of observations in a sample

xi, xgo « • * , xn

values in a sample of size n arranged in

increasing order

k number of order statistics used in estimating a

parameter

u any estimator of parameter u; also used for estimator

x - C p

a

p any estimator of parameter p

u& estimator of parameter u formed from k order statistics

f & estimator of parameter p formed from k order statistics

u* order statistic estimator of u when

p is unknown

Uq Gumbel^s estimator of u when p is unknown

v^ unbiased estimator of u formd from k order stotistics
5

E (T) expected value of any estimator T

cr 2

(T) or V(T) variance of T

Xi - n^/n, ratio of rank of order statistic to sample

size n

bias in equidistant method of spacing order statistics

in sample of N largest values

Bjq- bias in Gumbel’s plotting method for sample of N

largest values

Xj_ ^

iife. order statistic in sample of k values from

9 extreme-value distribution with

parameters a, u

order statistic in sample of k values from

9 extreme-talue distribution with

parameters 1, u * 0 v C Euler’s number, Q*57721566o . •

c

s

correction factor for unbiased estimate of dispersion parameter

standard deviation of observed sample values;

IV, EXTREME-VALUE PARAMETERS

The extreme-value distribution to be fitted to a sample

of data has the form

(1) F(x) = exp [ -e" a(l_u)

] = Probability that an observed value is < x, x This is the cumulative form of the distribution* The frequency, or density, form is given by the derivative

f (x) = F* (x) s a exp [ - a (x-u) - e~ a ( x*u ) ] * This function has the general shape shown in Figure 1, which

is scaled so that a = 1 and u = 0 o The function F(x)
6

represents the area under the curve f(x) to the lef of the

ordinate drawn at the point x»

Fitting the extreme-value distribution consist in

estimating the parameters a and u from the data in and, or

of .estimating one of these parameters when the othe is

known from previous data* These parameters have me nings

analogous to the standard deviation cr and mean (or t ode ) “p. in the case of the normal distribution

gU) . expt- ]

. a ffzrr 2

The quantity l/a 9 like cr

, is a scale parameter meas "ing

dispersion about the central value; u is a location parameter analogous to p, of the normal distribution, and lesignates the most probable value or mode. The mean of the

extreme-value distribution is

(2) E(x) - u + 2. , a

where C is Euler^s number, 0*57721566. ,

Since it ;\s more

natural to deal with l/a rather than we put

p * i /«

and consider p as the parameter to be estimated* p uid u

will also be referred to as extreme-value, or extra: -tl,

parameters*
V. ESTIMATION OF MODE u WHEN p IS KNOWN

Estimation of the mode in the case where the other

parameter, (3, is known, consists in choosing a specified k

of the sample values x1? x2 , • ••, Xq and computing their*

mean

/>%

_

= i 2^

E fri

xn

1

where is the estimator of u formed with the k order stat<

i sties Xq » xn0

* •••?

1 c. i£ the k values n^, n2 ,

The problem is how to select

nk

Three methods of selecting, or spacing, are considered

in this report: (i) "optimum" spacing, which gives a more

efficient estimator than any other method of spacing; (ii)

equidistant spacing, used in E. I. G-umbel f s graphical method

of analysis (reference 1, Lecture 2, pp. 20-21); and (iii) a

method of equating expected values analogous to one advocated

by B. F. Kimball (reference 5). These estimators are compared below with each other and with an estimator given by

Cxumbei which uses the mean of all sample values.

1. "Optimum" spacing . a. Use of one sample value . From the expression for

F(x) in equation (1) we see that the probability that a

value is less than the mode, x = u, is

F(u) = e

-1 = 0.36788 .
Hence in a sample of, say, 1,000, we would expect the

368th value in order of size to give a good estimate of

u, in general, for n large, our first estimator of the

mode u is

^1 = Xn

x

“ x ,368n

where

ui denotes an estimator of u using one sample value ; X 4 368n denotes the ordered sample value whose rank

is „368 times sample size n.

A

The quantity u^ is a statistic whose value varies

from sample to sample . Its distribution is approximately

normal for n large — the larger the n the closer the

approximation — with (asymptotically)

f

1 ]

(3) Mean U]_ = E(u]_) = u

(4) variance of ui = cr g (u ) * 1

1 n[f(u)3 s

where

X = F(u) = “

flu) - dF(x)1 B a s l0

v

' dx Jx«u e pe

Unless otherwise indicated, all formulas in this report

involving characteristics of estimators such as mean,

variance, bias, efficiency, will be understood to hold in

an asymptotic sense only, i.e® in the limit as sample

size n becomes indefinitely large. But for all finite n,

however large, the relationships are to be regarded as

merely approximate, the approximation being better (in

general) the larger the sample size.
9

Since the average value of u^in repeated sampling is u,

• A

the parameter which is estimated, we say that uq is

(asymptotically) an unbiased 3 estimator The variance

o^(ui) (also denoted by V(uq)

)

is

a

2 (up - ia^Up 8

.

In this relation the variance on the left measures the

degree of precision obtainable from a sample of n observations, and the equation shows that precision is inversely proportional to the sample size.

An inequality (due to Cramer and Rao, reference 7 ) in the theory of statistical estimation shows that under

certain general conditions the smallest variance (and

unbiased

hence greatest precision) obtainable with any^ estimator

of the parameter u, for a sample of size m, is not less

than

2,

A

° ^ u ) min il

m

From the above it is seen that when <r2(uq) 38 o

2

(u)

S « 33 0.5820o

n e - 1

This means that for a given degree of precision, the

theoretically smallest number of observations necessary

id* The term unbiased in statistics is generally reserved for

the case of complete absence of bias in samples of any

size, small or large. For simplicity, since the present

report deals only with large n, the teffcr is applied also

to estimators which are not strictly unbiased for finite

n, as long as their bias disappears when n increases

without limit.
10

(m = *5820 n) is not much over half the number (n) required by the given estimator u^. This condition is

true generally — the actual number of observations is

always greater than (or in some cases equal to) the

theoretically smallest number * The ratio m/n thus has

values only from 0 to 1, and may therefore bo used as

a measure of ”§fficiency*" It is more conveniently

calculated from the ratio cr^Cu)/ cr 2 (u^ ) formed for

m = n, and we write

(u) min

1

A

and say that the estimator ui is about 58 percent as

efficient as a theoretically most efficient estimator

Can a more efficient estimator be obtained? If in

A> place of 55 xn^ above, we use any other ordered sample value

(5) V =

then Hosteller (reference 3) has shown that

(6 ) E(ui f ) = u®

( 7 ) g

2

(u ' ) , h+lllkL

n(f(u3 )] 2

where the rank

n® = Xi n , 0 < X < 1

and u® is the position of the ordinate which cuts off

the area (on the left) under the curve f(x) 0
11

By varying ^1 it is found by trial that

x^ ~ «2o

gives the smallest variance and therefore the greatest

efficiency, 64.8 percent. Hence, theoretically,

^l' = z .20n

gives a "best" or "optimum” estimator when a single sam=

pie value is used.

However, if U]_ f is used to estimate u, a correction

must be added, since by equation (6), U]_ f estimates not u

but u* and is therefore biased . The value of the bias

is defined as the amount by which an estimator of a parameter over or underestimates it ( on the average L the

bias being positive or negative accordingly. Using the

fact that the area under f(x) up to the ordinate at x

is F(x), we have from the definition of u* , equation (1),

and the definition of X^

?

(8 ) F(u ?

) s exp [ -e“ a ( u9 ~u ) ] • X-^

? whence

(9) u* = u + — [-In (-In li) ]

a

- u - ( ,47588 $ , where In denotes logarithm to the base e. The quantity

in brackets may be obtained, for example, by looking * up the value X s * ,20 under the column headed ^ in

Table 2 of the National Bureau of Standards® forthcoming

probability tables for extreme values (reference 4),
Thus u which estimates u* , not u, requires a

knowledge of the other parameter p in order to estimate

u without bias. This disa&vantc: jx l . * 0.-1 '.iU u bhe

greater efficiency of over u^« The less efficient

estimator u^ can be used without knowledge of p* If p

is known, then the estimator to use is

* (10) v_ = x * .4759 p

.

1 .20n

This is unbiased and also most efficient among the n

order statistics. This result is listed in Table I,

columns (1) - (6), line 1.

b. Use of two or three sample values . If instead

of one order statistic we are willing to use k suitably

chosen ones out of the n, we can estimate the mode by using

(11) u. = 4 2 X . k 1*1 n^

The generalizations for equations (6), (7), and (9) are,

respectively,

k

(12) E(dk

) = £ 1

2

1

u

± ,

(13) a

2

(u.V) - I[ 2

n i-1 £f (» )]*

k Xid-x,)

t

i<J f(u

t

) f(^) J

(14) i A V • u XJ ] P } , where
13

For k * 2 it is found by trial that

* .08, Xg ® *40

give optimum spacing. After taking account of the bias,

given by the term in P in (14), the unbiased estimator

to use is

V2 = (L/2Xx s08n X o40n ) + .4074 p , with efficiency close to 82 percent as shown in Table X,

columns (1) - (6), line 2.

Likewise for k ® 3, the optimum estimator is

'3 * k/3^ x.05n + x o 20n + I.45n ) * * 4494 * ° For this statistic the efficiency rises to over 88 per-

- • - ;» o •; \

cent, as shown in Table I, columns (1) - (6), line -3.

%

Computation rapidly becomes more complicated for

higher values of k. If we were to continue till k * n

then all jtltffeple values would be drawn into the eSti-

^ yv tiS mator u^ = un . I.e., the estimator in question would

then be merely the sample mean corrected for bias,

Un

= u = x - Cp , with efficiency .6079, as shown in the last line of column (6) of Table I,, Hence if we were to compute optimum

estimators for increasing values of k, it is noteworthy

that efficiency would not always keep increasing. la

fact, we must ultimately reach the efficiency <>6079 when

all n sample values have been taken, which implies that

for some value of k, the efficiency would reach its marimum value and from then on, the efficiency would actually
14

become worse , as more sample values were drawn into the

estimator.

Example 1 . It is interesting to apply these modal

estimators to the data given in Example 1 of reference

1, which uses a different method. The data, consisting

of 485 maximum values of gust velocity, one maximum for

each of 485 traverses of thunderstorms, are tabulated

in Table II taken from that source. For rough comparison purposes, the value p « 1/a * 4.8263, found in the

example of reference 1, is assumed here. The results of

estimation by means of the optimum spacing procedures

of the present report are:

n = 485

k = 1; V! = X97 + (.4759) (4.8263) - 10.2258

+ 2.2968 = 12.5226 (assuming that linear

interpolation to find the 97th value is

valid)

1 k = 2: V2 * <5(x30 o g

+ *194) + ( .4074) (4.8263) *

1(7.7630 + 13.4483) + 1.9662 = 12.5719

k = 3: v3 = j(*24i#i5 + x97 + x218.25^ + (,*4494) *

(4.8263) = ^(6.4405 + 10.2258 + 14.3000)

+ 2.1689 = 12.4910 . These values are listed in Table III and compare very

well, considering the saving in computing labor, with
15

the value u = 12.8370, found in reference 1 by metes

of the formula u = x - — (derived by E. Jo G-umbel in

u

Lecture 3, page 18 of reference 2), which necessitates

calculating the mean of all 485 (grouped) values. The

"efficiency” (as here defined) of this latter estimator

of u (assuming l/a * 3 is known) given by the Grumbel

type of approach can be shown to be 60.8 percent — less

even than the 64.8 percent (Table I, column (6)) that

can be obtained by using only one ("optimum") sample

value

.

Table III shows that the "optimum" estimates obtained from the estimators v^ 9 y% 9 (column (2)),

although reasonably close to that given by Gumbel 9 s estimator u (column (5)), are all less than tu Ideally,

the various estimates should scatter among each other

in random fashion « The fact that they do not «ay be

attributed to grouping (see remark below) or to the

fact that the data depart from an underlying extreme®

value distribution. It is even possible, though very

unlikely, that the sample size n * 485 is not sufficiently large for the normality assumptions made above to

be sufficiently valid.

Remark. The closeness of one estimate to another

computed from the same data by a different estimator

should, of course, be regarded merely as suggestive,

rather than as a definite indication of a statistical
16

property of the estimators . The characteristics of estimators can he studied dependably only by theoretical methods

or by extensive experimental sampling * For large sample

sizes such as in the above example some weight can be given

to the numerical relations obtained, but it should always be

kept in mind that these might be upset through sampling variation if another sample of observations were taken*

It should also be noted that since the data are only

given in grouped form, interpolation is necessary in order

to obtain the desired order statistics* It is likely that

more satisfactory estimates would be obtained from data

given in full detail* Data should always be available in

their original observed form, before grouping or other processing is performed*

The above remark is relevant to all of the examples

given below*

2* Equidistant spacing . Beside optimum spacing, there are two other methods of

selecting k statistics in a sample which have been suggested

One of these, the simplest possible, is to divide the ranks

1, 2, **e, n into equal groups, i*e® by taking

M

= ITT! *

Thus, for k = 9, we select the 9 values

x 8 ln’ x *2n> • • • 9 x*9n>
IT

the values of which divide the n ranks into 10 equal groups 0 The equidistant method has the advantages that no extensive trial calculations are involved and that it is easier

to remember which values to pick out of a sample* Moreover,

the work of computing the estimates is as simple as finding

quart iles or percentiles 0 A disadvantage of the method is that it does not depend

upon the form of the population under observation* Exactly

the same numerical estimates will be obtained for the normal,

Type III, or practically any continuous population* Thus,

conceivably we may lose a warning signal which would ordina*

rily be provided by an estimator sensitive to the form of

population, since in that case the estimates would be out of

line with experience more frequently if the observed data did

not actually come from an extreme-value population than if

they did come from it*

Another reason for studying the method of equidistant

spacing is that it is intimately related to certain plotting

procedures advocated by E* J* Gumbel (reference 1, Lecture

2, pp* 20-21), as will be explained below*

The bias and efficiency of the equidistant method can

be obtained by the same procedures as used in optimum spacing, embodied in equations (11) to (15), with - i/(k+l) 9 The results for the first 5 values of k are shown in Table I,

columns (7) and (8) 0 The method overestimates the parameter

u by substantial amounts, depending upon the parameter p

!
18

(assumed known), and a negative correction (coltran (7)) is

necessary in each case, the smallest being (-®3665p) for k»l 0 It can be shown that for increasing k the correction tends

toward (-Cp), where G = 0* 577215 is Euler’s number®

Efficiency (column (8)) of the equidistant method is

relatively low, starting from just under one-half for one

order statistic (k = 1) and rising to about 63 percent for

estimation by five order statistics* This is not even as much

as that obtainable from one order statistic under optimum

spacing, which is 64®8 percent (column (6 ))«>

The relationship of the equidistant method to Gumbel’s

plotting method is as follows® Gumbel uses a special probability graph paper (Figure 2) and plots horizontally the

values in the sample of extreme observations, x^, x2 ,

. arranged in increasing size® For each x^ he plots the corresponding ratio i/(N + 1) =1 along a vertical non-uniform

probability scale® Parallel to this probability scale is a

uniform scale of the function

(16) yjL = -ln(-lnki) * The reasoning then proceeds as follows. If the data come

from a true extreme -value distribution, then the points

(xi, y^) should be situated closely about the straight line

(17

)

x = u + ^

= u + py , where u and p are the parameters of the underlying extremevalue distribution®
19

This follows from the definition of 'the extreme-value

distribution* The quantity X, the cumulative probability

associated with the observed variable x, is by equation (1)

related to x by

. _ e

~ a(x-u)

X = e

e

; i « e *

,

a(x-u) = -ln(-lnX) * Hence the corresponding y defined by (16) satisfies

y

= -ln(-lnX) = a(x-u)

X . e # , x = u+ ^

= u+py, a

if x is taken as the variable whose values we wish to predict 0 The method given for fitting the straight line (17) is

either by eye or by a modification of least squares which

allows the line to pass through the mean of the x*s, x^, and

the mean of the corresponding y*s, y^, computed for the observed sample values. This condition is given by the relat ion

(18) u = % - pyN

connecting the (estimates of f the-) parameters u, (3,

Reference to equation (14) shows that this (sample) mean

of the y*s, namely,.

(19) yN

= N ^

yi = N

C -lnt-lnX^)] = b^ , i=l i=l

X = 1

i N + 1

,
20

is precisely the quantity in curly brackets in (14), if N is

put for k9 The Yalu.es of this quantity hare already been

computed in connection with the equidistant method and are

giY@n as the coefficients of («g ) in Table I, column (T ), and.

designated by bjjo The quantity b^ 32 approaches Euler ? s

number, C, as N becomes infinite 0 The equidistant method is

thus essentially equivalent to that of Gumbel* s plotting po~

sitions, with the difference that sample size N for Gumbal

corresponds, not to sample size n for the equidistant method,

but to the number k of order statistics selected out of the ru

These considerations make it possible to find the bias

Gurnbel 8 s method considered as providing an estimator of

the mode u given by equation (18), namely,

u s % - pyN

« xN - (3% o The bias is

(20) Bjj = E (u) - u - E(xh

) - j3bI

- u ** (u + Cp) - (0

since the expected value of the sample mean is the same as the

population mean, namely,

(21) E(x) = u + CP o Example 2 o Using the same data as in example 1, we obtain the results derived In Table . XT0 values

compared with other estimates in Table I1I 0 It appears that

the estimates given by equal spacing differ from those given

by the sample mean by roughly the same order of magnitude (about

0 2 to 06 ) as do the estimates given by optimum spacing in Example

1 0 This time, however, the estimates are all greater than that
21

given by the sample mean, whereas in Example 1 they were all

less. (See also discussion and Remark under Example 1.

)

3 . Method of expe cted values . The last method of spacing which will be presented acts

as a compromise, combining some of the features of the other

two. This method, based on expected values, avoids the extensive computation of optimum spacing by making use of a

special table and, unlike the method of equal spacing, is

sensitive to the form of population from which the data are

assumed taken.

The method of expected values is as follows. The variate

values uj_ in (15) are chosen equal to the expected values of

the order statistics of a sample of size k from the twoparameter population of largest values. The relation (15)

then determines the spacing Xi»

If XjL'jj. is the i^ order statistic in a sample of k from

the extreme-value distribution with parameters u, (3, then its

expected value is

E < xi,k) = u + E (yi,k)P , where E (y^ k) is the expected value for the distribution whose

parameters are u = 0, (3 = 1,^0 The values of E(yi'k) have been

computed by the National Bureau of Standards in a table (reference 6) prepared at the suggestion of B« E* Kimball, for

’Taj'This follows from the fact that, by equation (1), if x

is the variate in the population F(x) - exp[ -e“ a( x~u)]

9 and y the variate in Eq(y) = exp(-e“-y), which is the

population fVwith parameters u = 0, (3 =1, thenx and y

are related byx = u + I = u+ y(3

,

a
22

i = 1(1 )n or 25, whichever is smaller, and n = 1(1)10(5)60

(10 )100 s . From the definition of the expected value method

we have

i

Ui - E(xijk ) = u + . For use presently we notice that from this relation and

equation (12) we have

s(S]d = 2 J

Ui = E( 2 4

Zi,k)

1=1 1=1

= E{xk ),

where is the ordinary mean of a sample of size k0 Since

the expected value of the sample mean is always the population mean, we have by equation (21)

(22) E(Qk) = E(x) » u + Cf3 . Returning to determination of X^, we have from equations

(1) 3 (15), and the relation a (3 = 1 , = F(ujJ = exp [ ~e~ ^ui“U)] ^ exp (-e“E ^i,k) ) 0 Since Efyi^'k) are known, the values X^ can be looked up in a

table of the extreme-value cumulative distribution exp (-©“7)

such as Table 1 of reference 4 a Determination of theX^ for the expected value method

is shown in Table ¥, and the biases and efficiencies of the

,xsethoc, in Table I, columns (9) and ( 10

)

0 If p is known,

then the method, for practical purposes, is unbiased, ^ince

tie bias, Cp , and the correction, -Cp = -0 a 5772p, are always

the same known values for any kG This follows ehsily from

equation (22), since the bias of in estimating u is, by

definition,
23

E(uk) - u = C(3

The efficiencies of the expected values method are even

lower than for equidistant spacing, starting from about 42

percent for estimation by a single order statistic and being

only 57 percent for estimation by five statistics.

The method of expected values bears the same relation

to a graphical method advocated by Kimball as does the equidistant method to the plotting procedure of Gumbel. Kimball ff s

method consists in modifying Gumbel's graphical procedure by

simply using for the values given by the method of expected

values instead of the simple ratios i/(k + 1), and leaving

everything else unchanged. The result is to replace equation

(19) by

, N , N V =1

±

2

=1 71 =1 2

=1 , which has been shown equal to C. Hence the bias in Kimball’s

method is

%* = (C - C)P = 0 ; i.e., his method has the theoretical merit of giving an unbiased position for the centroid of the fitted line.

Example 3 . Again using the data of Example 1, we perform

the calculations for the expected values method as in Table

IV, the results of which are listed in Table m for comparison.

It appears that this method gives estimates about 10 percent

closer to ti given by the sample mean than the equidistant

method gives. Again, the values are all higher than for the
24

sample mean0 (Compare Example !<>)

If one of these two systematic methods of spacing order

statistics is used, it appears that at least 3 should he

taken, since the last two examples indicate that k = 1 and

2 give relatively widely discrepant values as compared with

larger values of kD This precaution, however, does not seem

to he needed with optimum spacing 0

VI. ESTIMATION OF DISPERSION PARAMETER p

1 0 Use of two sample values . Hosteller (reference 3) has discussed the use of the

quasi-range

P2 “ fcn2

“ *

ni )/c

to estimate the standard deviation of a normal population

from a sample of n e This estimator may also he used in the

extreme-value case for estimating p, provided £ is chosen as

c = u2 - ui , where u^, U2 are defined by

F(uj.) = xii/n , i = 1, 2,

as in equation (15) fl Again using trial methods, we find the optimum spacing

to he given hy ^ =v *03, Xrg j= , with e

33 3 0 07159* Hence for two sample values, the optimum

estimator for p is

(23) 35 0.3256 (x o85n - X o03n ) ,
?5

and its efficiency ^I(as defined earlier) is 59 c 5 percent,,

A

Two points are worth noting about p 2 o First, it is unbiased,

bias being taken care of by the numerical factor 0 Secondly,

unlike the estimator for u, the estimator for p does not depend on a knowledge of the other parameter ue 2 0 Use of four sample values 0 The above result for two sample values can be improved

by taking two additional values* While exact determination

has not been carried out, a considerable amount of trial indicates that probably the best efficiency for four values is

reached by using the estimator

P 4 = 0.2026 (x. 85n + x. 70n - x a0n - x <03n ) c Its efficiency is 68*9 percent* As for the case of two

values, the numerical factor eliminates the bias, and use of

the estimator does not require a knowledge of the other

parameter u„

Example 4 . If we use the same data as in Example 1, the

above two estimators of p give the following results?

2 values: $ 2 = 0.3256(x412o25 “ x14.55)

= 0.3256 (21.4583 - 3.9182) = 5.7111

4 values: p 4

= 0.2026 (x412 . 25 -*• X330 . 5 S- x48.5 ~ x14.55)

= 0.2026(21,4583+ 18.4754- 8.2708-3,9182)

= 5.6211

M In the case where both parameters are unknown, the concept of efficiency defined earlier for the one-parameter

case is not strictly applicable* See section YII below

for further comments.
26

These values do not seem to be in as good agreement with the

value p = l/a * 4 C 82S3, found In reference 1 (by means of

jfcr

the formula l/a = - ° b given by Gumbel) as is the agreement

between the order-statistics estimates of u and the value of

u found in reference 1 by use of the sample meanc This is

not too surprising in view of the much greater reduction of

computing work through avoiding calculation of the standard

deviation as compared with the reduction by avoiding the

mean, ^ince ordinarily it is to be expected that more information is lost when the reduction in labor is greater, other

things being equal 0 For comparison, the efficiency of the Gumbel estimator

which is {asymptotically) unbiased, is approximately 39 percent o This value is obtained on the assumption that the

sample size n ® 485 is large enough so that the first one or

two terms in a Taylor series expansion of s furnish a good

approximation® This assumption has ’been found to give usable

results in similar situations 0 Determination of its validity

depends upon the exact evaluation of the variance of s 9 which

requires a prohibitive amount of numerical integration by

ordinary methods * C

D1

UJ Special sampling procedures in process of development

give promise of shortening the labor to within feasible

limit

s

0
27

VII. ESTIMATION OE MODE WHEN BOTH PARAMETERS ARE UNKNOWN

When both parameters are unknown, one might seek to try

a combination of the above methods. One would first estimate

p by the quasi-range with two or four order statistics and

then use this value in conjunction with one of the estimators

for u discussed previously. A complete discussion of this

method would, however, involve the question of joint estimation of two unknown parameters and is outside the scope of

this report. However, one simple case will be touched on

briefly.

We can try a combination of the most efficient single

sample value with a multiple of the difference of two of the

sample values. The estimator is, from (10) and (23),

This estimator is unbiased since P2 unbiased. Its variance is, by a modification of equation (13),

(24)

u* = x.20n + *47588132 = x .20n + « 15493 < x .85n - x.03n) •

+ 2 ( . 15493) [

X2

(l - Xg)

^

- x2

) ^ f f

2 3

1 3

$
28

where — .03, = ^3 = *85, and the other quantities

are as defined in (13) and (15) above. When evaluated, this

gives

Y(u) = 1.7423p 2/n . The corresponding Gumbel estimator of u, when (3 is unknown, is

& — •/&

*G * x “ %

Cs

where s is the sample standard deviation. As in the case of

A

{3, the Gumbel estimator of (3, the exact calculation of the

variance Uq. may involve prohibitive labor. The approximation

A

method used in connection with p gives

V{uG ) = 1.4042j3 2 /n . The concept of efficiency previously used, where only

one parameter was unknown, is not directly applicable to the

case of more than one unknown parameter, but involves the

concept of joint estimation mentioned above. Instead, we

may proceed as follows.

In the one-parameter case, 'two different unbiased estimators u^, U£, of one of the parameters, say u, were compared

by dividing the variance of each into a common value, namely,

the theoretically smallest value (under certain conditions)

which the variance of any (unbiased) estimator of u could

have. This furnished for each estimator an index, called

efficiency, of the form

eff (u ) i

g^(u) min

a

s(u ) i

9 i * 1, 2 a
29

Instead of evaluating each fraction separately, we could

A A

compare the efficiencies of u^ and U2 by simply taking their

ratio,

eff (ui) _o*2 (

u

2 ) eff ^

2 ) ~<r2(^

l)

say, 0 < k < 1,

A

and say that the estimator uq is k times as efficient as the

A

estimator u2 . In the two-parameter case we can also compare estimators,

in similar fashion, by taking the ratio of their variances, as

k above, and designating this ratio the "relative efficiency"

of uq relative to u2 , with the understanding that the word

"efficiency" will not necessarily have the same meaning as in

the one-parameter case.

We then have, subject to the approximations described

above in connection with the estimator $,

V(up) k = = 0*8060 e

That is, the estimator of u constructed from order statistics

is about 81 percent as "efficient" as the estimator given by

G-umbel, in the case where both parameters are unknown* Thus

the order statistic estimator u f can be used for rapid calculation without too great a loss of accuracy over that of the

estimator of Gumbel which requires computation of sums of

squares*

This discussion is for p unknown* If (3 is known, then u*

should be replaced by one of the more efficient estimators

listed in Table I*
30

Example 5 , Applying the estimator in (24) to the same

data as used before with (3 now unknown gives, from the results

of examples 1 and 4 above,

u f = 10.2258 + .47588 (5.7111)

= 12.9436

p 2

= 5.7111

.

The estimate of u is remarkably close to that obtained by

Gumbel's formula, 12.8370, but does not require the labor of

squaring, adding, and taking a square root. The estimate of

(3 is of course the same as in example 4.

VIII. CONCLUSIONS

Several large-sample unbiased estimators of the parameters of an extreme-value distribution have been developed

which, with one exception, appear to have greater efficiency

than those derived from methods of E. J 0 Gumbel and B. F.

Kimball and yet require much less effort in computation, involving essentially a mechanical sorting to find pre-designated ranked values in a sample of size n.

The unbiased estimators which can be recommended are the

following linear functions of order statistics, where, for

example, X 9 Qzn means the (-ilLj'kk observation in the sample

when all are arranged in ascending order, and t! @st. u” means

"estimator of u," the subscripts being omitted for simplicity.

Application to a sample of n = 485 maximum gust-velocity
31

observations analyzed in NACA TN 1926 (reference 1) has been

found to give satisfactory re suit s*

1. For estimating the mode u when (3 38 l/a is known ® Uses

(a) estoU = x e 20n + °4759 P® Efficiency = 65$

(b) est,u = ~(x e Q8n + x «40n) + ®4074p® Efficiency = 82$

(c) est. u = i(x o05n + x

.20n

+ x

,45n )

+ o 4494p » Efficiency = 88j$

By comparison, estimation by using the mean of all n values,

as in the Gumbel method, is about 61 percent efficient®

2* For estimating u when p is unknown . Use:

est 8 u = x

# 20n + ° 1549 (xo g5n

“ x

o 03n^ 0 This estimator is 81 percent as "efficient" as the estimator

given by Gumbel which involves the sample mean and standard

deviation, but avoids the work of squaring and summing®

3o For estimating 3 whether or not u is known. Use:

(a) estop - «3256 (x

o 85n ~ x o 03n^® Efficiency = 59^$

(b) est„(3 = .2026 (x. 85n + Xo70n

-x.10n " x.03n)* Efficiency = 69%

These a :ree less well with the estimator requiring the standard deviation, but the efficiency of the latter is indicated

to be about 39 percent®

Other methods of choosing the order statistics which

were examined may have some theoretical advantage and perhaps

be somewhat simpler, but they have substantially lower efficiency than those recommended®
32

Finally, it should be urged that when data are presented

for analysis, they should be given in their original detailed

form, before grouping or other processing. This would obviate the need for interpolation or other devices which may

tend to vitiate the accuracy of the analysis.

Statistical Engineering Laboratory

National Bureau of Standards

Washington, D. G.

15 August 1951
33

REFERENCES

lo Press, Harry i The Application of the .Statistical Theory

of Extreme Values to Gust-Load Problems 0 NACA TN

1926,' 1949 0

2„ Gurabel, E e J© s Lectures on the Statistical Theory of

Extreme Values and their Practical Applications®

Manuscript to be published by the National Bureau of

Standards

o

3® Mosteller, F»: On Some Useful n Inefficient” Statistics*

The Annals of Mathematical Statistics, vol* XVII, no« 4,

Dec* 1946, pp* 377-408®

4« National Bureau of Standards i Probability Tables for

the Analysis of Extreme-Value Data* (in press 0 )

5c Kimball, B* F a i Assignment of Frequencies to a Completely Ordered Set of Sample Data® Trans®, Am® Geophys*

Union, vol® 27, no® 6, 1946, pp® 843-846,

6 « Table of the First '"orient

to be submitted to IV CA

Standards,

)

of Ranked, Retro -. by the National Bu

r- K.‘ a&u cm

lo Cramer, Harald: Mathematical Methods of Statistics®

Princeton University Press (Princeton), 1946, p® 480,

equation (32«3o3a).

I

TABLE I

Comparison

of

three

types

of

methods

of

estimating

the

mode u

of

the

distribution

of

largest

values:

(i)

use

ofk

order

statistics

spaced

in

three

different

ways, p

(= A

34

bO fO . CT> X ' —

-pC3 *H £3 p ra o •

f-* & a © 3 a d d o

fH P ca OJ o 3 — • _ X

•H II <P © O £6 lx • <p ©dOXwp

t>> op©

•HO

•H

Vi

ViW©>

•HP©1—l ©

t>> 1 o

•h

©

Vi

©

Vi *H W O

oq

w® a r- P » b*

CtJ *H <D > bDH d © © U P © o© *H

boO©

Pi

CO

lx

©

rHPi 1 o

•H

©

Vi

©

Vi -H H O IUop ca 00 c\j •h n tci -p t- © o cacn

•h

© o • CQ f-i I I

to 1 o

•H

©

Vi©

Vi d nq o d X •H

o'©

Pi

co

•p

Pi o

Vi CVJ Vi b %

ro X

c\j X

rH xX

ca

CM caCMttID

ca

CM

C— ca

CM

c-H I ID

co cn t- o ID CD

ca

roo

ca

cn

ID ca

H-t-o

Mca

cn

M- o o

M- CM O 00 ID

CM O O

rH CM tO

01© 01©

•PP© -P © P

Fh

©

© *tH Pi O

•rH CD Vi . ^ O ©

CO t

1*9*

IX

«:?

ca

CM

tca

CMt-tCM I fc

ca

CM B t-C- caiD o •

bO

TABLE II

Distribution of n = 485 maximum gust velocities

Class interval

in ft. /sec.

Frequency

Cumulative

frequency from

smallest

2 to 4 4 4

4 to 6 11 15

6 to 8 27 42

8 to 10 48 90

10 to 12 62 152

12 to 14 58 210

14 to 16 55 265

16 to 18 60 325

18 to 20 61 386

20 to 22 36 422

22 to 24 17 439

24 to 26 18 457

26 to 28 n

O 465

28 to 30 7 472

30 to 32 6 478

32 to 34 3 481

34 to 36 1 482

36 to 38 2 484

38 to 40 1 485

485

Source: Table 2 of reference 1

(1946 Thunderstorm Project data)
(
TABLE III

Estimates of mode u obtained by various methods

from a sample of n = 485 extremes

p = 4.8263

k

(1) ...

for three methods of spacing

lc order statistics

Estimation

by

sample mean

u - x « CP

.(5)

"Optimum”

(2)

Equidistant

.(3)

Expected values

(4)

[ 1 12.5226 13.4130 13.6008

2 12.5719 13.1876 12.9896

3 12.4910 13.0767 12.9594

4 13.0114 12.9008

5 13.0031 12.9794

n = 485 • 12.8370

Source of basic data: NACA TN 1926, Table II
'

*
Calculations

for

estimating

mode u

from

sample

ofn=

485

extremes

37

COo

•H-P

CO •HP©P

co©©

'd©O

Lf) opw

•©op©a

CO©Pi—I ©>

^3©p0©

01©

fl •HO©

ft

CO I—I

cdP

cd9

ftO©

CO©

s» .a 0

c~ “T

© in

Ti 00 5 c- a ca • O CM CD co H- CO ft -—. O cn cn o cn o i i CM O 00 in o tp

rH cO cn cn cn cn

© ft ft

— • • 0 « • p <P <P ro CM CM CM CM

Gd rH rH rH

ii n •Hp •> in ft W <©

©' in K) rH in rH

fci -— cD in in CO in

cd i—1CO c- sr 00 co

CO

© ft rH IO 0- c- CO tp

©

© <P >—" • • • 0 • © f> CO LO in in LO 1—1 c rH rH rH rH •H

cd

_ „ in O CO CM ft O' ro o CM CO cn cm ft t- M-

© i Td -—-co cn ft oo cn tp c— cm cn t- co M- oo cn co p © M © © o CO cn in cn ro cn tp co ro co in in ro cm v o i—1 © ft © P rH ro ft ro in ip co co in in cn o cm in co ro © ft p a p d • 0 0 0 0 0 0 0 0 0 0 0 0 0 ft rH W Sh rH © Ho CO CM cn o in o cn ro c- ft © cm in © ro X Q 1 1 rH fH H H C\J H i—1 C\J ft ft ft CM w CO t> — ft © in in in o o o o o o in in o o in in •H in o ro ro co cm CO 00 lO CO CO r—1 C— M" M" x cn tp CM M- CM co lo cn tp ft co t- cn o m- a n cn in C- CM in oo ft CM CM ft T- ro ro cm cm M- © • • 0 000 0 0 0 0 0 0 0 0 0 lO t- CO 00 CP CM C\J O'— rH rH c- cn cm m- ro

•H c- in co o m o oo cn ft cm co in in ^ ro © CM ft ro ft CM ^ ft ro m- ft CM ro 'MrO ro ft OO co CM co co co cn cn co cm c— c~- o in in ro ft cn cn co ft oo oo oo o cn ro

CO e- cm in CM ro CM cOO'flO ro CM CM O cn

•H LO ro c- cm in oo ft M- CO 00 ft ro in c- oo X • o o o o o o o o o o o o o o o ©p © o CO t- M- »H

cd -— ro c- c0 rH ro a o c- 1—1

00 E— rH O

•a a 'H* rH O o O p 0 0 0 0 • CO ft ro ro ro ro ro o rH rH rH ft H o a —

ft »0 CP

©

©

© 00 ro CO CO o © P ft H CO rH 00 rH 3 o o ©—- © CO in cO in rH •h a © a CO c— cn o rH CM W P i-l cq. ft p • 0 0 • 0 © Op ft ft H rH rH CM CM CM ft ©rap © o 1 i 1 i 1 PQ © © l E-i o

ft ©

©

M 00 cn ro o

t»0 tit X Wjij rH 00 in ro .ao © m 00 ro H* CO 1—1 ©©

rH0 rH0 rH rH CM

cd > ft in in in in in

ft

CO <4<p rH rH rH rH rH

rH

cd •> i ft 00 ro v- ft oo cn co ro t- ft ft ro oo ^ ro 3 © ra © © ro cj- oo ft in in oo co ft oo ro ft M- cn a4 i—1 © ft © p H- 00 ro P- o oo M- CM M- CO ft ft ro oo m- o

ftp

© p © rH ro cn O ft CM CM M- CO ft co ro ft cn o Q H M © H # • 0 0 0 0 0 0 0 0 0 0 0 0 0 3 © HO in CM CP ft in cn o ro co o © cj in r- ft

CO > ' ft rH rH rH i—1 rH rH rH «H rH OJ rH rH rH OJ ©

•H CM r0.ro M- CM M- co ro cm ro co

A:© X \rH CM ft W\ft i—1 ro LO OJ «H rH rH

OC ii K>

ft CM ft ro ft cm ro tP M- ft C0 O ft CM ro m- •H CO CM CM M- cO cn cn cn co CD CO M" CM O © CM ft ro ft cm ro ft cm ro ft CM K) PrH

T- +

CM ro ro M- CM in in in in co ro cm ro co

ft CM \ ww

—>SNS\'\VS i rH ft CM rH rH ro ft cm ro V- rH rH rH OJ LO

•H X H rH CM ro in NOTE:

For

comparison,

the

estimate

ofu

given

by

the

sample

mean

isu=xC(3 »

12.8370
<
38

TABLE V

Calculation of spacing constants

for expected values method

k

A

E(yi>k )

i = 1 i = 2 i = 3 M- it

•H i = 5

1 0.5772

2 -0.1159 1.2704

3 -0.4036 0.4594 1.6758

4 -0.5735 0.1061 0.8128 1.9635

5 -0.6902 -0.1069 0.4256 1.0709 2.1867

k

= exp [-e -E(yi ,k)]

i

i

ii

•H CM II•H i = 3 i = 4

II CJI

1 0.5703

2 0.3253 0.7551

3 0.2238 0.5316 0.8292

4 0.1696 0.4068 0.6416 0.8689

5 0.1389 0.3286 0.5202 0.7097 0.8937

)) Source

See text, Section V, 3
: .xp ;
*)
f«x)

03

<£>
Developed

from

DrE

J

Gumbe/'s

Environmental

Protection

Section

\

Reseorch

and

Development

Branch

Extreme

Probability

Paper

, by

the

Climatology

Umt

Military

Planning

Division

,

Office

of

The

Quartermaster

General.

40

Figure 2 Specimen of extreme-value probability graph paper

THE NATIONAL BUREAU OF STANDARDS

Functions end Activities

The National Bureau of Standards is the principal agency of

the Federal Government for fundamental and appi ied research in physics, mathematics, chemistry, and engineering. Its activities range

from the determination of physical constants and properties of ma- terials, the development and maintenance of the national standards

of measurement in the physical sciences, and the development of

methods and instruments of measurement, to the development of special

devices for the military and civilian agencies of the Government.

The work includes basic and applied research, development, engineering, instrumentation, testing, evaluation, calibration services,

and various scientific and technical advisory services. A major

portion of tne MRS work is performed for other government agencies,

particularly the Department of Defense and the Atomic Energy Commission. The functions of the National Bureau of Standards are set

forth in the Act of Congress, March 3, 1Q01, as amended by Congress

in Public Law 619, 1950- The scope of activities is suggested in

the listing of divisions and sections on the inside of the front

cover .

Reports and Publications

The results of the Bureau’s work take the form of either actual

equipment and devices or published papers and reports. Reports are

issued to the sponsoring agency of a particular project or program.

Punished papers appear either in the Bureau’s own series of publications or -in the journals of professional and scientific societies.

The Bureau itself publishes three monthly periodicals, available

from the Government Printing Of fice : the Journal of Research, wmch

presents complete papers reporting technical investigations; the

Technical News Bulletin, which presents summary and preliminary re- ports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use r'or

radio communications throughout the world. There are also five

series of nonperiodical publications: the Applied Matnematics Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous Publications.

Information on the Bureau’s publications can be found in NBS

Circular 460, Publications of the National Bureau of Standards (75

cents). Information on calibration services and fees can be found

in NBS Circular 483, Testing by the National Bureau of Standards

(25 cents). Both are available from the Government Printing Of fice

.

Inquiries regarding the Bureau’s reports and publications should be

addressed to the Office of Scientific Publications, National Bureau

of Standards, Washington 25, D. C.
(