NATIONAL BUREAU OF STANDARDS REPORT

NBS Project NBS Report

1103-11*110? 1699

A MOTE ON NON-PARAMETRIC METHODS

Xo Riehard S&^ag©

<M$S>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issued fi production in any form,

writing is obtained froi Washington 25, D. C.

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

i, reprinting, or re*

i unless permission in Bureau of Standards,
t i ' f ' Vj
FORWGRD

‘Ails report, is a synopsis of a talk presented by Mr*

Savage entitled univariate aspects of multivariate

nonyparametric statistics” presented at the "Statistical

Engineering Seminar33 9 National Bureau of Standard®*, 16

November 195>3U

Work on this research project is carried out at the

National Bureau of Standards under the general supervision

of Churchill &isenh&rt e chief of fch© Statistical Englnoering Laboratory o Hh© Statistical Engineering Laboratory

is Section 11 «3 of th© National Applied Mathematics Laboratories

(Division ll p National Bureau of S tandards) and is concerned

with to© development and application ©f mode m statistical

methods in toe physical sciences and engineering

o

Jo Ho Curtiss

Chiefs National Applied

Mathematic® Laboratories

Ao Vo As tin

Acting Director

National Bureau of Standards
- ' • >v2 J

... *
A NOTE OIT NON~PARAMETRIC METHODS

By

I o

ivi chard Savage

Hati©nai Bureau of Standards

SUMMARY

The main tools of non-param© trie statistics &r© described*.

It is pointed out that some of these tools d© not hav@

natural extensions t© multivariate problems*. The possibility

of finding tests of randomness that are independent of

ordinary tests of hypotheses is explained*,

I INTRODUCTION

The object of non-parame trie methods is to make probability statements about random variables when n littl@'n

’ is

known of the distribution of these random variables 0 It Is

hard to give th© precise meaning ©f w littie5" in the above

statement© It refers her© to the fact that all we know about

th© random variables of interest can be phrased in terms ©f

such general function theory concepts as continuity of th©

distribution functions*, the existence of a density function*,

th© possibility of factoring the distribution functions that

are involved 9 etc©

It is interesting to note that most of the work that

has been done in a non-parana trie manner has been concerned

with real-valued (scalar) random variables*, Exceptions to
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this are® the work done on multidimensional tolerance limit®

[2] and rank correlation [$]o We shall point out that fehls

is a natural limitation of the common tools that have been

used for most of this work*, i 0 s o th© use of th© probability

integral transform*, ©f topological invariance*, of randomisation

and ©f dish© torn! &&ti on P IX PROBABILITY INTEGRAL TRANSFORM

The probability integral transform technique says that

if Ffx) is the cumulative distribution function (continuous)

of th© random variable X*, then th© new random variable Y »

F(X) has a uniform distribution.? This fact has been most

useful In achieving by non®parametric methods such results

as finding confidence intervals for th© median [15

3

» finding

confidence Intervals for a distribution by th© W&Xd-Wolfowifcs

method [11]* developing the Nsyinan smooth test [9 Jo ©too

Simpson [10] pointed out that the corresponding transformation for two-dimensional random variables 8 and consequently for higher dimensions does not have the desired

properties 0 However*, in facing a multivariate non-parometric

problem it Is sometimes possible t® consider a real-valued

random variable that is a function of th© several original

random variables*, and in this manner the problem can be reduced

to one that can be handled o This is essentially what Simpson

did in [10j o

Mo Rosenblatt in his paper ('’Limit theorems associated

with variants of the von Mises statistic” presented at the
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March 21 P 1952 contributed paper session ©f th® Slst meeting

of th© Institute of Mathematical Statist!©©) 8 has gi won m

analogue to th© probability integral transformation which

should prove useful in future mu! tivari ate p non-parametrie

studieso

III TOFOTjOOTCAL invariance

Tf points are placed on a line* and then a topological

transformation of th® lino into itself is mad©*, th© order

of th© points on th© line either remains th© same or is reversed

Many non-parametric procedures depend on this fact*, ic@ 0 th©

Wald-Wol fowl fcz test of goodness =©f~fi t using the total number

of runs [ 12

]

0 the Mann test of randomness [73© the MannWhitney two-sample test [83c ©fcco

Wolfcults [16] pointed out that for higher dimension©

there was no analogue to this procedure 0 This is most unfortunate since it has turned out that this is one of th©

most useful of the non-param© trie method©

a

2t is interesting to note that rank order statistics

and symmetric statistics are independent*, given a sample

of real-valued continuous random variables that are identically

and independently distributed [U3* By a rank order statistic

is meant a statistic which depends only on the order in which

the ranked variables were observed*, and thus using the

topological Invariance p its distribution under random sampling
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is non^par&m© tri c <? Examples ©r such a?© the longest run

above the median and the statistics used by Mann |7]

fc@ test for trendo By a symmetric statistic is meant a

syjnmetric function of the sample observation® 0 The median*,

the mean*, ®tco 0 of the sample etc*, are symmetric statistics,:

The proof ©f the independence theorem for rank order statistics

and symmetric statistics and its extension to several samples

is simple „ The independence ©f rank order statistics and

symmetric statistics is useful in situations where we wish

to test the hypothesis that we have a random sample from a

population with some specific property; for example B th©

hypothesis that w® have a random sample from a population with

median equal to sero« Ordinarily in this type of testing

problem all of th© emphasis is put on testing whether th©

median is equal to zero; but in many situations it would

also be Important to test whether the sample was randoms

This method allows us to test both ports of th© hypothesis

with tests which are independent of each other when th© null

hypothesis is true* Thus it is possible to get known significance

levels [lij]o

IV RANDOMIZATION

The method of randomization is based on the fact that

under the assumption of random sampling we can find the exact

conditional distribution for any statistic once we have made

the set of observations [12 ]<>
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In this method there Is rio problem in extending feh©

technique to the multivariate case© Her© the main problem

consists in the excessive computation required© llius we

would hav© to cake a separata table giving th® distribution

for each sample drawn© It does not seem possible t© give

a simple technique for making exact probability statements

using this method^ although the asymptotic theory has been

worked out in detail T 3 3

for th© univariat© ease©

V DXCHTOKIZATTON

Many problems where non^parame tri e methods are called

for can be so transformed that all of the variables involved

have binomial distributions© Typical ©f such problems is

the testing of whether the median of a continuous distribution

the

is equal to zero© Here one classifies/observations as positive

or negative! then one proceeds by using the fact that the

number of positive and the number of negative observations

have a binomial distribution©

By changing one®s problems so that they only involve

binomial variates one quickly has the required distribution

theory© This method has been used by Lehmann [6] for attacks

ing multivariate problems© Perhaps the biggest drawback in

using this method is that for some problems it appears t© be

very inefficient [6]©
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lo Batsmen p Go On the power function ©f the longest pun as

a test for randomness in a sequence of alternatives*

Bi©m8trika3 3$ (1948)°

2 0 Fraser, Do A© & Q Sequentially determined statistically

equivalent blocks© American Mathematical S@©i@tyP 22 (1951). Po 372 © - ' “

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3© Hoeffdings, A combinatorial central limit th©@r©m«,

Annals of Mathematical Statistics *, 22 (1951) Po

558," " n® © %T J

4o H© telling p Ho and Pabst 0 Mo R© &ank correlation and

tests of significance involving n® assumption of

n®rmality0 Annals @f Mathematical Statistics*, ?

(1936) po Zlo

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5o Kendall 9 Mo ®o ^ank Correlation Methods ., Charles Griffin

Company Lirif

_t®^7""Lond®n 0 1946"o

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6o Lehmann 0 E© Consistency and unbiasedness of certain non=>

parametric testo Annals of Mathematical Statistics,,

22 (19515 Po 165«

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7o Mann p He. On a test for randomness based on signs of

difference© Annals of Mathematical Statistics,, 16

(19455 8 Po 19To

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8o Marm 0 H© and Whitney^ D© Ro On a test of whether one

of two random variables is stochastically larger

than the other© Annals ©f Mathematical Statistics^

18 (1947) Po 50o

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9© Neyro©n„ Jo ni>mooth Test” for goodness of fit© Skandlnavisk

Aktuarie tidskrl

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e (1937) Po 149o

10o Simpson,, P c Note on the estimation of a bivariate distri=

bution function© Annals of Mathematical Statistics*,

22 (1951) Po 476 o

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11 o Wald s A0 and Vfolfowifcz J© Confidence limits for continuous

distribution functions© Annals of Mathematical

Statistics ,, 10 (1939) po TBITS

12 o Waid 0 A 0 and Wolfowitz, Jo On a test whether two samples

are from the same population 0 Annals of Mathematical

Statistics g XI (19iiO) po l47o

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Statistics fl 15 (I9kk) No. ko ————————

11*0 Wallis ff Compounding probabilities from independent

significance t©sts Q Jbj©onom®tri©a 0 10 (1942) No. 3o

15 o Wilks e Order statistics© Bulletin of the American

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