NATIONAL BUREAU OF STANDARDS REPORT

2116

THE VERIFICATION OF AN HYPOTHESIS

CONCERNING THE NORMALITY OF DISTRIBUTIONS BY SMALL SAMPLES

by

A. A , Petrov

Translated from the Russian by Curtis D, Benster

Editor: D. Teichroew

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1101-11- £100 December 10, 19 £2 2116

THE VERIFICATION OF AN HYPOTHESIS

CONCERNING THE NORMALITY OF DISTRIBUTIONS BY SMALL SAMPLES*

by

A. A. Petrov

Translated from the Russian by Curtis D. Benster**

Editor: D. Teichroew

*D0KLADY AKADEMII NAUK SSSR 19 £1. vol. LXXVI, No. 3, pp. 3££~3£8

s|c j|c This translation was sponsored in part by the

Office of Naval Research

<^b|>

The publication, rep Approved for public release by the

In part, Is prohibited

unless permission Is Director of the National Institute of

tandards, Washington

25, D, C, Such pern Standards and Technology (NIST) sport has been speclfically prepared If tha on October 9, 2015 port for Its own use,

THE VESICATION OF AN HYPOTHESIS

CONCERNING THE NORMALITY OF DISTRIBUTIONS BY SMALL SAMPLES*

** ***

by

A. A. Petrov

1. Let there be given an aggregate of N samples of size n %1 > > *

1*1 *

9 9 0 0 0

0 0 0 0 9

X

N1» ***

> x

Nn *

The question is raised: is the assumption admissible that the i-th

sample (for any i) has been obtained by a random selection from an

infinite aggregate having a cumulative distribution function

F(a^x + b^), where F is a given function, the same for all samples,

and the parameters a^ and b^ may be different for different samples

and are unknown to us. (This assumption will henceforth be called

hypothesis F.) In particular, is the assumption admissible that

each of our samples has been taken from a normally distributed

aggregate with mathematical expectations and dispersions different

for the different samples.

*DOKLADY AKADEMII NAUK SSSR 1951. vol. LXXVI, No. 3, pp. 359-358

Reviewed in Mathematical Reviews 1951, 12:622 .

(Presented by Acad. A. N. Kolmogorov, Nov. 29, 19 50)

^ ^ This translation was sponsored in part by the Office of

****

Naval Research

2

For verification of the stated hypothesis it is natural to

consider some functions or other, 7j (x^, x^, , x

n

), the distributions of which, computed on the assumption that x^, •••

, x

n

are independent and subject to the distribution function F(ax + b),

do not depend on a and b. One can consider, for instance, the quantities

7

xmax

s

x

. 7

.. x . - x *' min

x. - x

i

whose distribution in the case of a normal law F has been studied

by N„ Vo Smirnov Another method, starting from a distribution,

9 ) first considered by Thompson , of the quantities 7j ^ has been proposed by Arley and Buch . The method proposed below

has the advantage of not requiring the computation of the [root] mean

squares s. In addition, it gives n - 2 separate curves for comparison

and may therefore be a more powerful means of discriminating the types

of distribution,

<)

2. Let X be a variate having a distribution of continuous type

with distribution function F and probability density f = F' and let

W See

< 2 > See

^

See

^

See

^

See

[ 1 ].

[ 2 ].

[3l o

[Id.

[*].
. 1 ' ' . -
3

x^ - x^ - s • ° - x^ be a sequence of n independent observations of

the variate X, arranged in increasing order. Let us consider the

ratio

K - \ *1

(where 1 < k < n)

These quantities are invariant with respect to the choice of the

scale and the origin of coordinates , and their distribution functions will therefore be identical for all distribution functions

F(ax + b) differing from F(x) only by the values of the parameters

a and b.

It can be shown that the distribution function of the variate

4k

yn - a)

ni

(k-2 ) I (n-k-1) 7 Jff [F(y) - F(x)] k

~ 2 [F(z) - F(y)]

n' k

' 1 X

X f (x) f (y ) f(z) dxdydz , 1,0

0.8

0.6

o.U

0.2

/>

r

/

/

/ 4

/ J

" Jf

>r

a s b /

/ > y

//V

f

tf

1 V

o.U o„6

Fig. 1. F

2 (t)

0 0.2 0.8 1.0

a

where the integration is carried out over the region G defined by

the inequalities

x - y - z, <t .

3. Let the quantity X be distributed uniformly on the segment

(0, 1). Then, on the strength of formula (1) ?

(2) Vt) = (k. g) , ( ":k.1) T J'jJ

' (y - x)

k' 2

( Z - y)

n-k"1dxdydz ,

G*

where the integration is carried out over the region G* defined by

the inequalities

0-x-y-z-l ~

J — < t .

* z - x

From the foregoing we obtain, in the case n = £, for example:

F

2 (t) = 1 - (1 - t) 3

, F

3

(t) = t

2

(3 - 2t), F

u

(t) = t

3

.

U. Let the quantity X have a continuous and strictly mono'

tonic distribution function F. By means of the substitution

X' = F(X) the distribution F is reduced to a uniform one, and the

distribution function F^(t) is defined by integral (2) over the

region

r~

1

(y) - F

-1

(x)

FX

(.) - F'bx)

9

Fig. 2 „ F

3

(t)

where F is the function inverse to F„ Carrying out the integration in expression (2) with respect to the variable x, we
£ -
6

arrive at the following formula, which is suitable for numerical

integration:

(3) F

k

(t) =

nj

(k-1 )

1

(n-k-l)l

/ vn-k-1/ \k"*l 3 (z - y) (y - x

o

) dy ,

where

r'hy) - tF~

1

(z) 1 e 1 - t

The recommended method of verifying the admissibility of

hypothesis F consists in the following. Each of our N samples

gives an observed value of each of the quantities [l<k<n] .

By these N observed values, empirical distribution functions are

constructed for the quantities . By means of Kolmogorov's criterion^, a comparison is made between the empirical distribution

functions obtained and the theoretical, which have been computed

previously by formula (1) or (3). If significant deviations are

thereby revealed, be it for even one k, the hypothesis F under

test is rejected. In case no such deviations are discovered for

ary k at all, the data contained in cur samples agree well with

the hypothesis F.

x = x (y, z, t) = F

o o

* *

( 1 ) See [6] .
.

.

,

'*

.

.
7

6. For the practical application of the method described, the

functions F^ corresponding to the given value of n and the hypothesis

under test, F, must be known. The computations were conducted for

n = 5 and for two hypotheses—the normal and the uniform distributions.

On the accompanying graphs are exhibited the functions F^, F F^,

computed for uniform (b ) and normal (a) distributions by formulas (2)

and (3), and the empirical distribution functions of the quantities

(^2* 3* k

} cons-kructed in accordance with a sample of 200 groups

of 5 taken from tables of random numbers subject to the normal law^.

In all three cases the empirical distribution functions agree well

with the hypothesis of a normal distribution. On comparison with

the hypothesis of uniform distribution in the twQ cases and<^)

we obtain significant deviations. These deviations correspond in

the first case to a level of significance of 0.06% and in the second,

to a level of significance of 2%.

In conclusion I utilize this opportunity to express to A. N.

Kolmogorov my heartfelt gratitude for having posed the problem and for

direction during the fulfilment of this work.

<«See [ 7 ].

Received Nov. 27 , 1950

December 10 , 1952

Fig. 3. F^(t) on next page
'

"f

.• :

'' "' v

'

' ; v . . : .. ;

• i

- 'v: " . i.

'

' l

.

v. .

.

5

.

'

•

' • :

.. . 1 :

•

" ' '

' 1

;

••

. j ,

i
8

- A

/

p

ftf /

f

'

;

ftA/

4

-b

/

.ft

/ j / ft/

ftr /

-

—g^ft

. — •! 0 0.2 0 .U 0.6 0.8 1.0

Fig. 3. F, (t)

9

BIBLIOGRAPHY

[1] N. V. Smirnov , [On the estimation of the maximal member in a series of observations], Doklady Akademii Hank, 33, 3h&, (19 hi).

[2] W. R. Thompson, [On a criterion for the rejection of observations

and the distribution of the ratio of deviation to sample

standard deviations], Ann. Math. Statistics, 6, 21 h (1938)*

C3l N. Arley, [On the distribution of relative errors from a normal

population of errors], K. Danske Vid. Selsk. Mat,-fys. Mead.,

18, No. 3 (I9h0).

[

Ll] N. Arley and K. R. Buch, Introdxiction to the theory of

probability and statistics, 1980.

[8] H. Cramer, The mathematical methods of statistics, 19 U6.

[6] V. I. Romanovskii, Primeneniia matematicheskoi statistiki v opytnom dele [Applications of mathematical statistics in

experimental work] 19 h7.

[7] H. Wold, Random normal deviates. Tracts for computers.

No. 28, 19U8.
. . I

v .
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