Section 11.3 RLE COPY

Do not remove NATIONAL BUREAU OF STANDARDS REPORT

1621

EMPIRICAL SAMPLING DISTRIBUTIONS II

fcy

D. Teichroew

Institute of Statistics

University of North Carolina,

and

National Bureau of Standards

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

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1621

EMPIRICAL SAMPLING DISTRIBUTIONS II

1' 2

tor

D. Teichroew

Institute of Statistics, University of North Carolina

and

National Bureau of Standards

PREPRINT

'"Address delivered at the meeting of the Institute at Boston,

Massachusetts on December 26, 1951*

2This report was completed under a National Bureau of Standards

contract with the University of North Carolina*

• • j Approved for public release by the

Tftia report is issued r in any form, either in Director of the National Institute of £ram the Office of the Standards and Technology (NIST) on October 9, 2015

on, reprinting, or reproduction

rmiaaion in writing ia obtained

ashingtan 25, D- C.

MBS

k-h-$Z

PREPRINT

1 2 Empirical Sampling Distributions. II 5

by

D 0 Teichroew

Institute of Statistics, University of North Carolina

and

National Bureau of Standards

1. Instroduetion

2. Statement of Problem and Notation

3 « Sampling Models

k» Estimates of H(s)

5® Comparison of Variance of Estimators

6 * C-omputation

7o Applications

8o Numerical Example

9

c

Summary

^Address delivered at the meeting of the Institute at Boston,

Massachusetts on December 26, 1951

2This report was completed under a National Bureau of Standards 1 contract with the University of North Carolina e

1* INTRODUCTION

The calculation of distribution functions is one of

important tasks in mathematical statistics o Of particular

importance are the sampling distributions of statistics used in

testing hypotheses® These require the determination of the

distribution of a function of5 say ns random variables,, each of which

has the same basic distribution® The sampling distribution can

be expressed in the form of a definite integral and5 if this

integral can be evaluated analytically5 it is only necessary to

prepare a table of the resulting function, if not,, the table

may be obtained by numerical integration®

An empirical method of determining the sampling distribution

suggests itself immediately from the. formulation of the problem®

This method consists of taking,, say N5 samples,, each consisting

of n observations on the random variable and calculating the

value of the statistic for each sample® The "erpirical"

distribution can then be obtained by arranging the values of the

statistic in order of size and counting the proportion that

are less than^ or equal to a given value® (In practice5 it is

usually desirable to apply a smoothing process to the resulting

distribution)

®

The empirical method has been used since it was introduced

by Student [6] 5 without the indication that it could be considered

as a way of getting approximate values for a definite integral®

In recent years9 much attention lias been directed towards the

numerical solution of difficult mathematical problems® It has

3

been found that probability interpretations may be used^ not only

to evaluate definite integrals,, but also to invert matrices and

to solve differential equations© The general class of methods

has been given the code name of Monte Carlo and now has a

fairly extensive literature9 (see for example,, Curtiss [2] s Kahn [ 3

]

9 and The Monte Carlo Method [8]) c

An outstanding contribution of Monte Carlo theory is the

use of so-called "importance sampling"© The concept will be

developed in Section 39 the word "importance however,, will be

avoided since it is not particularly appropriate © As far as we

know^ no one has yet utilized this concept in calculating sampling

distributions© In this papery some of the theoretical aspects

of such a procedure will be examined,, with particular attention

to the practical aspects of using the method on high-speed

electronic calculators©

It might be mentioned here that empirical sampling^, because

of probability considerations^ cannot^ with a practical number

of samples,, give results which are accurate to many significant

figures© Even with the fastest electronic calculators now being

built5 statisticans* by virtue of economic necessity,, may have

to be satisfied with thousands of sanples instead of millions©

Nevertheless^ there are many practical problems which could be

solved b3^ results obtainable most economically by empirical

sampling© For example ^ one such problem is to determine which

of a number of non=parametric tests is most powerful against a

h

specified class of alternatives © A problem of this type is being

prepared for computation by the SWAC (National Bureau of Standards

Western Automatic Computer)

©

2 0 STATEMENT OF PROBLEM AND NOTATION

The basic model consists of j

(i) A random variable x which has a density function f(x)

(ii) A sample size n

(iii) A statistic s which is a function of n independent

observations on x»

In any practical problem,, various values of n and various functional forms of s may arise* and this may affect the method of solution

but for the present only one value of n and one form of s will be

used©

The problem is to calculate (and tabulate) the sampling

distribution of s* i 0 e 0 the probability that the statistic will

be less than or equal to some given value© Let H(s) denote this

function of s* then

where the integration is taken over the region in the n dimensional

sample space in which

In accordance with standard usage* capital letters xd.ll indicate observed values* a dash beneath a letter denotes a vector*

and the n dimensional space of possible sample values will be

H(s) " Prob
'

-
5

denoted by R « In addition f(x) will be written for f(x, 5 x « x

dx for dx5

®

e ® dx^, a single integral sign will be used for nfold integration. The range of summation will be indicated by the

subscript,, summation over individuals in a sample will be for j

from 1 to n and summation over samples will be for i from 1 to N.

In addition it will be convenient to defines

A(s) s s s(x) 4 sj

~ set in R in which s(x) ~ s, n —

[x) 3 1 if s(x) ~ s

“ 0 otherwise

i.e if x C A(s)

i.e if x c P“ r “ ^-( s

'

( 2 . 2 )

(2.3)

g(x) " any n dimensional density function

i.e g(x) ^ 0 foi all x in R j

v(x) "

(2.U)

R: g(x) dx 3 1

n

w(x)

f(x)

g(x)

With this notation (2.1) becomes

H(s) f ( x) dx - t(x) f(x) dx

(2.3)

( 2 . 6 )

(2.7)

n

3. SAMPLING MODELS

Equation (2.7) can be regarded either as giving the -value of

H(s) as a definite integral or as stating that if a random variable

x has a density function f(x) then %

E £t(x)J = H(s)

since the expectation operator is defined by just such an equation

(Cramer [1], p. 170) » This means that the average value of t(X) in

N observations on x will tend to H(s) as N increases, i.e. the variance

or the dispersion of these average values around H(s) is a decreasing

6

function of N. Therefore one way of estimating H(s) is to take many

samples (according to the basic model) and calculate the average value

of t(X).

For this case the behaviour of the estimator is well-known (and

is furthermore independent of f(x) ) since t(X.) is a binomial variate \

3- ©0 v

t(X) as i with probability H(s)

- 0 with probability 1 - H(s)

and therefore if follows that the estimator

P1

^ 38 N

2 t(x

±

) (3.1)

E

(

Pl ( s)J ® H(s) (3.2)

N Var pn (s) H(S') - [H(s )

]

2

(3.3)

This sampling model has consisted of two steps?

lo The space is divided into two mutually exclusive sets of points,,

A(s) and - A(s)»

2* Points are selected from the probability of a point coming

from A(s) is H(s) and the probability of it coming from - A(s) is

1 - H(s.)« An estimate of H(s) therefore is the proportion of points

coming from A(s)* The important point here is that the first part,

is a matter of geometry only and is independent of probability

considerations »

From a consideration of step 29 it is clear that the probability

of a point coming from A(s) does not have to be H(s) and it might be

possible to reduce the variance of the estimate of H(s) by changing

it* The mechanics of the procedure are apparent if the integrand

in (2e7) is multiplied and divided by a density function g(x)<> The

equation becomes

H(s) “ Jb “7*4 g(x) * n g(x)

(where v(x) is defined by (2.5)) and is interpreted as stating that

E \ v(x

where x has a density function g(x) c Nows in step 2 S the probability

that a point comes from A(s) is

s ) g(x) dx

and an estimate of H(s) is the average of v(X^) e The basis of "importance” sampling is to choose a g(x)^ such that regions in which have

a large contribution to the variance of the estimate are sampled

more heavily than other regions (since the variance is a decreasing

function of N) and therefore the overall variance of the estimate

of H(s) will be reduced*

The word '’importance** will not be used in this paper5 instead

the two sampling models will be referred to as Model I and Model II®

Both of these models will give an empirical answer to the problem

outlined in section 2 and differ in the distribution of the random

variable which is sampled* These distributions are as follows

s

Model Ig The random variable sampled has density f(x) e

Model II g The random variable sampled has density g(x) 4 f(x) <>

Ilag g(x) g(x

i

) g(x

2

) coe g(x )

lib g g(x) is not factorable as in Ila.®

Any statement made about Model II without referring to a or b5 applies

v(x) g(x) dx (3.U)

to both,

8

It may be noted that Model II sampling bears a certain analogy

to the stratified sampling used in surveys 0 A survey^ in its simplest

form,, consists of selecting a sample of m individuals from a population

of M individuals and observing the value of a characteristic on each

of the selected individuals o The average value obtained in this

way is an estimate of the average of that characteristic for the

population o If the population can be divided into k strata,, in each

of which the characteristic has a smaller variance than in the whole

population,, an estimate of the mean with smaller variance can some=

times be obtained by allocating different numbers of samples to the

different strata e Formulas can be determined for the optimum sample

sizes for each strata on the basis of strata variances and number

of individuals in each strata®

In Model II sampling the population consists of the points x of

the n dimensional sample space and the characteristic is s(x) 0 A

strata consists of points which have their values of s in a certain

intervale The allocation of samples to the different strata is

accomplished by choosing a g(x) e

lu ESTIMATES OF H(s)

The problem,, under Model II sampling might be looked at as a

selection of a g(x) 5 such that the variance of some estimator of

H(s) is a minimumio The minimium turns out to be zero 5 as has been

shown by Kahn [3]o This trivial solution arises because the problem

is 5 incorrectly^ being considered as one in statistical inference 0

In statistical inference a sample is taken from a distribution which

is a function of an unknown parameter and it is desired to estimate
-LSfcl
9

this unknown value from information obtained from the sample e In the

present problem the distribution from which a sample is taken is

completely specified and from this s ample an "estimate” of H(s) is

desiredo However* H(s) is not unknown because its value is given by

the integral (2ol)« Quite naturally,, therefore* the trivial solution

states that an estimate of H(s) with zero variance can be obtained

by evaluating the integrate The word "estimate" will be used in this

paper in the special sense that the unknown quantity is not a parameter

of the distribution which is being sampled,,

For sampling under Model I* the estimator (3d) is the only

reasonable one to use, Its variance (3*3) and* in fact* its sampling

distribution are known and do not depend on the functional form of

f(x) 0 Under Model II the problem is to estimate the expected value

of v(x) e This is a much more complex problem because the form of the

distribution of v(x) may depend* not only on g(x)* but also on f(x) c

and furthermore* this distribution may be such that the variance may

not be a suitable criterion of desirability „ Therefore the choice

of estimator will depend on the particular situation,, For the general

discussion* the variance will be used and two classes of estimators

having general applicability will be considered* first those depending on

the sample values and second* those depending on the ordered observations only*

Of the first class* the most natural one is the sample average*

which has already been mentioned in the last sections Let

Hence* the average value may not be a desirable estimator of E

This estimator has the unfortunate property that p^Cs'") where

s* is the largest observed value of s* is in general not equal to

10

unity, i 0 e c P2 (s) is not a distribution function in the usual sense e

The estimator may be normalized to make it run from zero to one by

dividing by p^s*')© This leads us to consider the "ratio*1 estimator g> 2 v(Xi) -

f \ > v

p

3

(s) —(D 5

(U.2)

where w(x) is defined by (2 ©6)©

Under the class of estimators depending on the ordered values

of v(X^) ^ estimators such as the median^ etc©,, might be considered,,

The problem of forcing the empirical sampling distribution to cover

the interval from zero to one would arise here too 0 Investigation

of this class is of secondary importance^ because the time and

memory space required to rank the s(X^) increases too rapidly as N

Increases to make the use of these order statistics attractive for

high speed computing machines

©

5o COMPARISON OF VARIANCE OF ESTIMATORS

p^(s) obtained under Model I has expectation and variance as given

by (3o2) and (3 ©3) respectively © For any particular value of s5 confidence limits for H(s) may be calculated by exact binomial theory

if N is small or the asymptotic normality of p^(s) if N is large©

Confidence limits for the entire function H(s) of the form

p1

(s) - 2 N

2 < H(s)c p? (s) + A N

3

may be obtained by the method given by Kolmogorov [U] if N is large

or by tables given by Massey [$] if N is small 0 Wald and Wolfowitz [7]

have given a method for calculating confidence limits of other forms©

11

p0 (s) is defined by (iui)„ Since

and var v(x) A

E fv(*)} s H(s)

[f(x)] 2

[g(x)]

1 dx -

[ H( s )

]

(5.1)

it follows that

- H(s)

N var Pg ( s

[f(x)]^ [ g( x) ] 1 dx - [H(s)]‘

(5.2)

(5.3)

e

c e Var,, Pp(s) - 0 If

g(x) * i ( x) [ H( s ) ] 1

if xC A(s) (5.U) e 0 otherwise

This is the zero variance solution for g(x) mentioned in section ho

It is not of much use in actual problems because it requires a

knowledge of H(s)j applies to only one value of s and furthermore

is defined in terms of the set A(s)„ If Model Ila were used5 where

gU) - 7Tg( Xj ) , g(x) could not be defined. In general, in ten*

of A(s)j, unless the statistic were of a similar form^ i 8 e e

s(x) - TTs(x ).

Next consider the more modest requirement of choosing g(x) so

that var p^(s) > var p^s),, i®e«

f (x) d2E > Sx

1

° (5.5)

A sufficient condition for this inequality is

g(x) > f(x) for all x € A(s) „ (5.6)

This however is not a necessary condition

12

Since p^(s) is a ratio of two random variables v(x) and w(x)

its distribution,, or even its ejected value and variance cannot9 in general,, be calculated explicitly,, The best we can do is to follow

the procedure usually used in a situation of this kind and consider

the expansion of a function of two variable

s

5 v s and w in a Taylor

sides of the equation gives the expected value of the ratio© A

similar series can be obtained for the variance^ etc, If an upper

bound can be found for the sum of the series beginning with the (m+l)th

term the first m terms can be used as an approximation with a known

maximum error. This is difficult to do in the general case but in

any particular situation it may be possible®

In the general case,, if the terms are decreasing functions of

N* as N becomes large,, all terms except the first in the series will

become negligible. These statements are made precise in the theorem

given by Cramer [1] p e 355 and 366© Here we need only the expected

value and variance of which the first terms are

series about the point E(v) 5 E(w)„ Taking expected values of both

E p^(s) ~ H

x „-l

(5o7)

var

p^

( s ) N

It will be noticed that N var v is the var p? (s) therefore

p^(s) can have a lower variance than p^s) only if cov (v(x), w(x))

is positive. Since

cov (v(x), w(x)) [f(x)] 2

[ g( x) ] 1 dx - H(s) (5.9)

var p^(s) < var p0 (s) only if

(5.10)

13

This is the opposite inequality to the one given in (£•£) Tor

var Pp(s) < var p^(s)„ The two methods of estimation are to some

extent complementary,

6, COMPUTATION

Before considering some of the applications of Model II sampling

it may be worthwhile to outline the changes in computational procedure that would be necessary. The computations carried out by

a high speed machine in computing an empirical sampling distribution

under Model I may be divided into three steps t

1, The generation and testing of random numbers and the transformation of these random numbers to observations from f(x),

2, The calculation of the statistic from n of these observations,

3, The preparation of an empirical distribution of N values of the

statistic. This could take the form of a printed list of the N

values (in order of size,, if possible) or a printed frequency

distribution^ each frequency being a count of the number of values

that fell in a certain interval (the intervals having been determined in advance).

Under Model II<, step 1 would contain the transformation of random

numbers to observations from g(x)

»

Step 2 would be unchanged.

However^ in step 3S it is now necessary to record not only the value

of the statistic but also the value of v(X) for each sample. The

empirical distribution could take the form of a printed list of

N values (in order of size^, if possible) together with the value of

v(X) for that sample or a printed distribution,, each frequency this

time being the sum of the v(X) c s for all, samples having the value

of the statistic fall in that interval*

The major change arises in the computation of the variance of

the estimates,, Under Model I, the var p (s), may be estimated

from p (s) (1 - p^(s)) for which nothing is required except p^

itself „ Var p^Cs ) however must be estimated from the variance

of the v(x) 8 s and if this cannot be done by evaluating the integral

(5 el) theoretically, it will be necessary to store the sum of [v(X)]

for all samples having values falling in each interval in the

frequency distribution,, p0 (s) presents an even more difficult

problem since^in general, no exact forraula for its variance exists „

One estimate of var

p^

( s

)

can be obtained by substituting estimates

from the sample in (5o8) 0 Another estimate can be obtained bydividing the N samples into k groups of N samples each, estimating

p^ from each group and calculating the variance of the k estimates,,

If the computations are carried out on IBM machines, the

calculation of v(X) may be time-consuming. Prof . J. ¥„ Tukey has

suggested that the work could be simplified by dividing the

x axis into a finite number of intervals and in each one choosing

g(x) so that [f(x)] [g(x)] 13 a

1 where a is some constant and

i - 0, + 1, + 2, + etc o Then v(X) can be calculated for each

sa.mple merely by summing the i

8 s that appear in that sample and

raising a to that power 0 This procedure greatly simplifies the

calculations but leads to estimates with large variance unless a

is close to unity®

15

? „ APPLICATIONS

It will be advantageous to use Model II sampling only if it

reduces the total cost 3 where the cost may be determined either by

trying to estimate H(s) with minimum cost for a certain amount of

precision or with maximum precision for a given cost 0 Since a

problem of this type is relatively easy to program for a high speed

computer^ the major expense in an empirical sampling study will be

the cost of time of the computer <, The amount of computing time

may be decreased either by decreasing the number of samples required

or by decreasing the number of computations required for each

sample ©

If Model I is used the number of samples required can be

determined by fixing the value of II(s) and var p^(s)© There

may exist a g(x) such that if it were used with Model II var p^Cs)

or var p^(s) would be substantially lower and hence the number of

samples required would be sufficiently smaller to more than offset

the increased amount of computing 0 Another case in which Model II

could reduce the number the samples is where the sampling is being

carried out mainly to determine the "tails'® of the distribution

of s.1000 samples under Model I will not give much information about

the value of sqfor which H(s) - o9999» Under Model II a g(x)

could be selected such that5 sa

y

5 an average of 10 out of every

1000 samples would have s ^ s^ and hence provide an estimate of

H(s

o

)„

Model II sampling could also be used to decrease the computing

time required for each sample „ The first step in the computation

16

consists of transforming uniform variates to observations from f(x) 0

In some cases this involves considerable computation and there may

exist a g(x) such that it is quicker to transform the uniform variates

to observations from g(x) and compute v (x) e Sometimes the distribution of a statistic is required for a set of functions

f^(x) 5 f^(x) 5 °°° o Instead of using each of these in Model I

sampling it may be advisable to select one of them as g(x) and

use Model II,, particularly if it is desirable and/or permissable

to use the same set of uniform variates for each of the functions 0

Under this procedure,, the value of the statistic would, have to

be calculated only once for each sample instead of once for each f^(x)

and this could be a great saving,, if5 for example the calculation

of the statistic involved the inversion of a matrixc

In using Model II sampling,, it is therefore necessary to

determine g(x) in a way that will achieve the desired effect a

Unfortunately this is a difficult tiling to do. Equation (5*5)

and (5.10) give conditions under which estimates under Model II

sampling may have a lower variance than estimates under Model I*

As methods of determining in advance which g(x) should be used

they are not much help since the integrals involved will generally

be as difficult to evaluate as the original one„ Therefore the

answer must come largely from intuition and empirical work and in

difficult cases mainly from the latter e

8 c NUMERICAL EXAMPLE

The following example has been chosen because the expressions

for the variances of p2 (s) and p^(s) can be evaluated analytically e

17

Suppose the sampling distribution of the sum of square of n standardized normal deviates is required and that Model II. sampling will be

used with g(x) a normal distribution with mean zero and variance

2

<y~ e Table I and II give the variance of p^ and the asymptotic

2 variance of p^ for different c?~ and H(s)| for n - 1 and 9

respectively 0 (The numbers given are actually N times the variance),

These values were calculated as follows

1

f(x)

g(x)

’X

2

/2

1 -X

2

/2 G-2

Cr V2T

p

* (ct-ct- 1 )

2 / "2 -X /2 or

*

O' V2lr

where cr

*2 o~ 2<y

2-l

[f(x)]‘

g(x)

(o~ a~

V n

n

VcrV2^

~2x . / 2 o-‘ e j

'

s(x) - 2x,

A(s)

- ^R

n

: s < s HJ

where H - H(s) Frob c ) s < sH

f(x) dx

A(s)

K (s)ds n

where K (s) is the Chi-Square distribution with n degrees of

freedom® Therefore

[ f (x)

]

2

[ g(x) ] 1 dx « {cr cr ) n

A(s)

K (s) ds

n' '

and this integral can be evaluated directly from table of the ChiSquare distribution®.

18

The variance under Model I is given by cr It may be

noticed that if o~ < 1^ p^Cs) will sometimes give a smaller

2 variance 5 if cr j \} p^(s) will sometimes give a smaller variance,

For n - 9 and H = o 80 all the variances are given to indicate how

rapidly the variances can increase if an inappropriate estimator

is usedo

April 1$9 1952'

19

TABLE I

Comparison of Variances of Estimators

under Models I and II, for n = 1

H - o

CO « H - .90 H - .95

2

o

~ Var Var" Var P^ Var" PQ Var P^ Var" p^

o.95 .01+8 •Oij.7 .0871

o.6o .085 .072 .0929

0.70 .107 .060 .0)419

0.80 .117 •06U .0378

0.90 .138 .075 .0385

1.00 .160 .160 .090 .090 .Oh75 .0575

1.25 .139 .068 .0318

i.5o .131 .060 .0252

2:.oo .127 .052 .0197

3.00 .133 .050 .0171

U.oo .051 .0167

5.oo a O .0169

"Asymptotic Variance

20

TABLE II

Comparison of Variances of Estimators

under Models I and II, for n = 9

H - o 80 H - •90 H - •95

2

s/ x/_ o~ Var Var ' P^ Var P

2

Var P^ Var P^ Var P,

0„60 068U 8 o 222 lo306 2*218

0o7$ •177 •597 • 265 •371

0 o 80 •135 o391 •159 •191

0o90 oil 8 0 222 •057 •059

IcOO Ol6o c l60 o090 •090 o 0U8 • 0U8

1.25 • 610 »136 •059 •021;

x«5o •9h7 •119 •0U6 •017

2 o 00 2 o 896 •2lU •062 •018

“Asymptotic Variance

21

REFERENCES

1. Cramer* (I9 I4.6 ) Mathematical Methods of Statistics*

Princeton University Press®

2 . Jo Ho Curtiss* (19U9) '•Sampling methods applied to differential and difference equations .

Proco of Sem® on Scientific Computation* IBM* New York* p® 87-109

®

3. Ho Kahn* (19U9) "Modification of the Monte Carlo

method."'* Proc. of Sem. on

Scientific Computing* IBM*

New York* p® 20-27®

U* A. Kolmogorov* (19lil) "Confidence limits for an unknown

distribution function®" Annals of

Math® Stat®* V® 12* p. U61-3

5® F® J. Massey* (1950) "A note on the estimation of a

distribution function by confidence

limits." Annals of Math. Stat®

V® 21* p. 116-9 .

6 . Student (1908) "The probable error of a mean"*

Biometricka V. 6 , p. 1

7 • A Wald and J, Wolfowitzr "Confidence limits for continuous

(1939) distribution functions"* Annals

of Math® Stat® V® 10* p® 105-18.

8 . Monte Carlo Method (1951) National Bureau of Standards.

Applied Mathematics Series No® 120

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