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NATIONAL BUREAU OF STANDARDS REPORT

1694

£> 1 MULTA HE OUS :SSTS OF RANDOMNESS

AND OTHER HYPOTHESES

By

I* Richard Savage

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Charles Sawyer, Secretary

NATIONAL BUREAU OF STANDARDS

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

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-21-511

9

23 May 1952 1694

SIMULTANEOUS TESTS OF RANDOMNESS

AND OTHER HYPOTHESES

by

I, Richard Savage

Research supported in part by the

U. S. Naval Ordnance Test Station, Inyokern

The publication

unless permissi

25. D. C. Sue

prepared if th

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

or In part, Is prohibited

f Standards, Washington

ort has been specifically

report for its own use.
FOREWORD

It is shown in this report that rank order statistics

and symmetric statistics are independently distributed under

the assumption of random sampling. Techniques are given

that are particularly useful in testing certain hypotheses

concerning stochastic processes.

This research was carried out in the Statistical Engineering Laboratory, and was sponsored in part by the U. S.

Naval Ordnance Test Station, Inyokern, under Contract

NPJ*5 -507/14-3 with the National Bureau of Standards. The

Statistical Engineering Laboratory is Section 11*3 of the

National Applied Mathematics Laboratories (Division 11 of the

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

Ao V. As tin

Acting Director

National Bureau of Standards
SIMULTANEOUS TESTS OF FABDOMNESS

AND OTHER HYPO DIESES

Bf

I o Richard Savage

Summarys In this report it is shown that under fch® hypothesis

of -randoi&aess rank order statistics and trie statistics

are independent© Tills fact i© ©f use in t© s ting . hypo these s

as is shown by ©samples©

In troduc ti on : A common problem in statistical inference Is

"Do the observations x.»»*o Inx corn® from a population centered

at the origin?” On® solution to this problem is to us®

th© ordinary "t" test of this hypothesis 0 A more careful

observer might notice that there Is no assumption of normality

which is needed in order to justify th© us© of the "t” test»

The observer might then Interpret "centered at th® origin"

as "median at th© origin" and use th® sign t@st 9 Finally

the- observer may notice that there is no assumption- that

the observations w©re made ©n independently and idsntiealy

distributed random variables? The objective of this report

is fc© show that many of th© tests of randomness (identical

su'd independent random variables) are independent of th©

common tests of hypotheses Q More precisely it will b©

shown that if one has random samples then many of the tests
v a; B'Jf. rw m -

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of randomness will be independent of ordinary tests ©f

hypothesise. Or under the null hypothesis of random sampling

tests of randomness ar® independent ©f ordinary tests ©f

hypotheses* Using this fact th© statistician will b© abl©

to test both th© ’"Fandom” and TO p&ramsfcrie” parts ©f a

hypothesis with known significance levels*

As a further example on© might hare x(t) an observation

on a Fundamental Kandom Process [Maim (1951^ ] V

e On© knows

that if th© null hypothesis is tru© then th© quantities

x(10) - x C(i»l)5] ^isl 9 o«on) form a random sample from

a normal distribution with mean gar® and variance proportional

to 5 o To test this hypothesis one must test for both randomness

and normal! ty Q %® teat for randomness would depend on the

alternatives of Interests perhaps ©n© of th© mm tests

[Levon© (19^2) 3 would b© found suitable* Th© test of normality

could bo performed using th© classical chi square g©odn@ss

of fit test with one parameter estimated',* If this w©r©

th© test program this report will show that th© run test

and th© chi square test ar© independent under th© null

hypothesis* •

Assumptions and Notation ? W© shall b© concerned with real

valued random variables or x± j (depending if there is on©

©r several samples p x being th© observation in - a*

o

Names' Tsl1 owed by dates in square brackets refer© to items

in the bibliography at the end of this report*
0» -»•'•«* iB® : ’
sample, and being th© observation In the i

lSil

stable)* W® shall assume that all of the observations

corn© from distributions that have continuous cummulativ©

distribution functlona a Letting P(A) stand for the probability

of the ©v@nt A(P(A|B) will stand for feh® conditional probability

of the ©vent A given that the ©vent B has ®©curr©d) 0 the

assumption of continuity implies §

Cl) P(si j*a) » ©

(2) p(j^ p

p ©nd either i f i« or j f j

# ) « 0

A a tatl stic 1© any measureabl© function ©f the observations

Cx^jlg, and will b© generally denoted t(x£|) A In som© of

Ll

th© applications th© statistic® used in this report will

actually b© vector valued^, and consequently all ©f th© proofs ai9®

given in th© case where t(x ) is a vector® In writing that

t J

a statistic is equal to a certain value or less than or

equal t® a certain value this should be interpreted In. terms

of the components of the statistic^® In defining certain

statistics subsequently we will not include any values if

th© ©vent in (2) (this event will b© denoted by E) above

occurs 8 ^inc© th© ©vent & has probability sero 0 any statistic

which is not defined explicitly when this ©vent occurs may

b© defined arbitrarily*, without effect th© distribution

of th© statistic®

The rank of n^jCin the i sample) Is a statistic

JLlU ©nd equals the number of observations in the sample"
; . h&- ss -ic ; -

• '> Vrc-:r.: rf ,o

d'l .

‘ *

ffl

' %Ji

J rf i >jj . . a
“ 4

CT

whos© values ai»© less than of equal to that of Xj-jo

A rank order statistic Is a statistic which la a function

only of the ranks (within samples) of the observations, and

will be denoted by Ho

A symmetric statist!© Is a symmetric function of the

observations within samples , and will b© denoted by So

She "t" statistic for two samples ii m example of

a symmetric statistic** If w© had several samples and from

©ash of them formed a run statistic ELevvne{l952) ]» then

any function of th© several run statistics would b© a rank

order statistic* In particular for @n@ sample th© total

number of runs up and down Is a rank order statistic*,

Lemmas 5 In the following lemmas w© shall need the sets

A.p defined as follows i (x^) belongs to Aj If

x

5^

< x $\°° a< Hb@p® ( h> Jn )

is a permutation of th© first n integers^ Then it is clear

that there ar© n® sets Aj® and that they are all disjoint®

Further th© sum ©f th© Bets A and th© set corresponding

to th® ©vent E[ t'eee (2) above J &r® disjoint and together

form a n=>dim©nsi©nal Euclidean space E

r0

Lemma Is P['R®rI (x^) belongs to Aj j

is ®n© or &©r@ depending
• ©tf lB0ps *xo a%d$ as©!

i ~fS**e a 8 *

*

. ; - t^mrs^a -3 ssi feljfr !> • ?.

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. .'•• .i no* Iiwd'yaii «ifc* to «©&*

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’ ?l-i?3*£ & ®1' iTvJOfe PF£^T

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.•is %-orS$ $£&$ tm a

- A Ir ©'xe ®#S£LVi'

5 r:;j , A 8i 8B 8

^ • - • s •'-'-

..&• aoii-q.a n®©b.lls£?a 5 l

- :t©J :i©as

'-

: :
5

on If there is at least on© point sfx'j) in A.| such that

R (%Q .| ) ® r or note

Proof s If (xjjand (x 0 ^) &r© two points in A ^ then it is

clear that and have the sam© ranks for each i

§*

2,000, Do But a rank order statistic is a function of tha

ranks only, and therefor© any rank order statistic can asauna

only on© value in the s©fc Ajo Prom this the lemma follows o Lemma II? Under the assumption that (x^) is a random

sample fro® a continuous distribution:

PCCx*) belong® to A, 3 ® l/n®

Proof : Let F(x) be th© distribution function* then?

P[|ki

) belong

[Let F(xf

) * y3 3

® l/n 0 Qo E 0 Do

O
3*0-, n> » ' %\ > 0

OU O ov-' o
6

Lemma Ills Under th© assumption of sampling from a distribution

with a continuous distribution fun@ii@ng

P£S < slfx^) belongs to A $

] s P(S < s}..: o

Proof g Let Sj be the set ©f points which ar© in Aj and stseh

, / that S < 3n Then it is clear that?

P{sj) P C o

j

'o )

H©nc© using Lemma II we .have that:

P[S % s|Cxj-) belongs to Ajj

PfS f s and C^|b©l©ngs to Aj]

ITf^5T

=B©I@I55^5'^^

s

Pfsj^ n« - *L e p(s

"T j

S

p C s < ») Qo Ko D o Fundamental Result : In this section w© ©tat© and prove

a theorem wh5 eh is immediately useful for th© problem about

th© Fundamental Random Process mentioned in th® introduc tionn

Theorem I? In samples of fixed siged of random

& variables (x^ ) which are independently and identically dis^

fcributed any rank order statistic and ©ay symmetric statistic

are independently distributedo
d (( | q

8 Ir ‘•'1 •

? ts”

.

'I6£« 8.’ " *
**• 7 *=*

Proof ? Clearly any particular rank ©pdar statistic R

c&n take on at most n 9 values and therefore has a dig ©pet© Q

dis tribute ©Bo And thus w® perform th© following analysis to

show th© independence of R and Ss

P(R « p and S < s)

ca * P[R * r and S < s and (x^) belongs to E]

<# p[R s p ©nd S < s and (x. ) does not belong to E]

a PfR » r and S < s and (x^) belongs to the sus of th© Aj°e]

s g p[R » r and S < s and (x ) belongs A 3 j

i i “ 2 PfS « s|r £ and (x ) belongs ts A.J

j

<=> ^ 1 J P[R sb r and (x.

)

belongs to AJ

1 J

(Now by Lesaraa I)

s SFfS «g s| (x») belongs to A J Q

J

** J P[R gs p and (x*

)

belongs t© Aj]

(Now by L©E®a III)

«* PCS < a) X PfR r and (x.) belongs t@ A^]

j

® P(S < s)P(R = p) Q 0 ISo So
C I %d l

sgnc led i z) ba*.”.-**

1 o’d p
G^> 8 CO

Applications of Tha©r@m I s (1) W© wish to test the hypothesis

that observations Sp 6 ,„x g

Croad© in that order) ©om© from

a distribution with median g©r@ ? Th® p@ss5.bl© alternatives

&r© that the median is greater than sero 0 or- that the median

is shifting toward larger values as the successive observations

are mad© 0 To test this hypothesis it would be appropriate to

us© the sign test statistic [MacStewart (1941) h> and the [Mann

(1945)1 statistic for trendo It is clear that large values

of either ©f these statistics should b® used to reject the

null hypothesise Th© sign test statistic is a symmetric

statistic and th© Mann statistic is a rank order statistic

so clearly Theorem I is applicable © In order to make a test ©f significance at th© a level 9 using these two statistics^ there sr© available two standard

techniques o W© may use th© x technique for combining

testa of significances An ex tensive analysis ©f this method

can b© found in [Tallis ( 1942)1 see also [Fisher ( 1925 )]*

®i© other technique is to ehoos© levels of significance

and ©g for th© two tests based on the two statistics*. Then

th© probability of rejecting th© null hypothesis (when true)

by at least on© of th© test is 1 - (l»a^) (l-ag) 88

- ®j_®2 o &Inc© and are at our disposal w© can usually

pick a desired combination of them so that ai a2

° al®2

“ a Q These techniques for combining independent tests can
. s’ *x©l [ . ; /:)

tfee ; eit& ; o 4 sol v y..; 40 ©IS , © to r D "©©flia 0

.

-
9

<?s>

of ©ours© to® used in all ©f the remaining example®,, When

p there are more than two independent tests the x method is

still appli cable » and in the other technique we pick levels

©f significance (snj there are e independent

tests) o Then the probability that the null hypothesis will

©

be rejected (when true) is 1 TT(l~®j_) and w© ©an usually

i»l

e choose the a, such that a 1 - TTU-a, » . 1=1

(2) W© v<?Ish to test feh© hypothesis that x( t) is an

observation from a fundamental random process 9 and the

alternatives an© that x(16) - x[ (1=1)63 Ci^l® oo ag n) bf® ©liber

not normally distributed or are autocorrelatedo Th© alternative

of not being normally distributed may foe test®4‘by using ‘a X g@odness=@f=fi t test on the variables x(i©) - z[(i->l)0] 9 after estimating the eommon variance of th© random variable®

x{16) ® x[(i-l)5]o 'This is a symmetric statist!©*, Th©

autocorrelation may be tested by using th© rank order

autocorrelation coefficients [see Noether (195®-)]° These

are clearly rank order statistics.. Thus Theorem I assures

2

us that under the null hypothesis th® % statistic is

independent of th© rank order autocorrelation coefficients^

Several Samples s In this section we shall state and prove

a theorem analagous to Theorem I ff applicable to several samples 0
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10

Th®oreai II z Assume that the set of random variables

('.&i jH * o p Mjocg^H) are independently distributed

and identically distributed for fixed i (itb sample^ then any

rank order statistic of the N samples is independent of any

symmetric statistic of the N samples^

N

Proof s The sample space is again the ^uclidaan n(n ® % a? ) 1*1

space E

n

E

a

can be written as the product ©f N Euclidean

spaces of dimension that is

U

E - TT En

i-1 i

Further each can be decomposed into sets E

x

& A>^

a

Ci”lp o o o^Ng, 3

o o o p h| 0 ) the sets E

1 and At » being analogous 4 ‘ to the sets K and A » of the lemmas,-,

E’ * TTC 2 A.1 * E

1

) B 3

Thus

which can be written asf

TT^'

kh Bk * E * Her© E is the union ©f all of th@s© sets in as expressed

in the above product which contain at least on© E'H That is

the set of

E is/th©se points in with at least two coordinates equal

n

N

within & sampler. Bj^ar© th© Tf 0 disjoint sets of the

N

form TT A, i

1^1 <?! » that is no ties within samples 0
J

“ ; .• . y-.s

:

ic *

'

5

i

•• v

•* * , : , n ' : ? t

• f

> * ' i

. . ) .

*

•j *•

!
• 11

It Is clear by the continuity assumptions that?

P{K') * Qo

Than s PCS « s and R « r)

s 2PfS < s and R ® r and (x« A belongs to B.J ^

" * j K

P|S < s and F? s r and (x.$ j) belongs t© S] « EPC S < s and B s f> and (x^ .J belongs fc© k

® SP|S < 8 1 (Xq J belongs to B^end R * -p-J k

P[ (xi belongs to and R s r]

PCs 5 s) P(K » r) Qo Eo d«

Applications of Theorem s Cl) Assume w© have an observation

on & stochastic process and wish to test the hypothesis that

the increments ©p© independent and identically distributed 0 The alternatives we wish to guard against are such things'

as autocorrelation and that xCt) has some marked change at

isome time t [

to ay x(t) is displacement in the horizontal ©£ a

shell and is the approximate time at which the shell

stops rising and start® descending]® Let ® x(

- x[t..*(i»l)G] and i ® *(n»4)»o.#»n o Then the null

at == ^ hypothesis implies that y

17 ^ » y^f i^Cn^-dL*, o <> o 9 0 3 and

y^ (i*l*

o

3 from two samples of Identical and independent
f

,

MX
observations*, We right use the [Kolmogoraff (19Ul)3 goodnessof~fit test to see that the observations from both samples

come from, the same distribution o lie t the statistic used b

®

called We can test autocorrelation by computing statistics

R® and R which are the rank crdei* autocorrelation coefficients

from the fix# st and second samples reapec tiveljo The vector

(R|H) corresponds to the quantity R ©f the theorem® and since

S is symmetric w© have under the null hypothesis that (R 0 „R ' )) and S ar® independent *> Also of course under the null hypothesis

R 9 and H are independents

(2) n litters of animals Cn^in th© ith litter) ©re to

b© used for an experiment*, Preliminary to the experiment w®

wish t© test that average weights of the litters are the same-.

The mil hypothesis is that all ©f th© weights &r© independent®

normal with same mean and variance <> As alternatives we shall

be eoreerned if there i s a birtb order effect within litters

or that th© litters have different average weights 0 The P

test cf analysis ©f variance will test the homogeneous nature

of the means® and th® total*'-

1

, number of rune up and down in

th® ith litter will be a tost ©f birth order * It is clear

that the F is & symmetric and the vector CFh) is a rank order

statistic and therefor© fch© theorem is applicable*.
, :

:

• "

C. ..

‘

- ‘

i *

'

.

' link; , ~ •. i

• r ei 1

t i-w bm '& s t

-

o;> vj,:

- 1 bih : area; >
<=? 13

Power i It is hard to describe the power functions of tests

based on several statistics, when the severs! statistics

do not regain independent^ When the testa are independents

even under the alternative hypotheses it is possible to find

the power function for each test separately and then of coups--*

it is possible to find the_ -power function in most eases fit

is trivial if we combine the feast results by the second method

discussed above-),,.

A f the random variables do remain independent and

Identically distributed® then even though the parame tri©

portion of th© null hypothesis may be false® th® statistics

used to feast the “random34 end "parame trie” parts of th© null

hypothesis do remain independentgeneral, iiatione § Th© results ©f this- report may be extended

to fch© ©ass where the observations instead of being rea5U

valued ars simply points in any abstract spac@o In this case

th© definition of symmetric statistic is not affected 0 However® rank order statistics must b© r@defin©d 0 On® way

of doing this is to form a real valued functional of th©

observations® and define the rank order statistics in terms

of the values of this functional « If this is dona in such

a manner that the set E still remain of measure &©r© then

th© theorems of th© report will be applicable t© this more

general system-
di od , -'.t-L ( . v d •

‘

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d a ieeridfow

to*&adq*bi ‘

ald«:virqq« id HI** ^cqi"'
BIBLIOGRAPHY

Bo A0 Fisher

1925 Statistical methods for research workers 0 Oliver and Edinburghj> Howard LeVene

• 195-2 On the power function of tasfcs of randomness

based on runs up and down«

Annals of Mathematical Statistic a 23 .9 34=56 «

**« MaeS tewart

1941 A note on the power function of the sign testo

Annals of Mathematical Statistics 12 s 236^239?

Henry Bo Mann

1945 On a tost for randomness based on signs of difference

Annals of Mathematical Statistics 16 *, 193=199?

1951 Introduction to the theory of stochastic processes

{'..©pending on a continuous oarametCFo

National Bureau of Standards Report^ 1293 *• Go t tfried E 0 N@e thcr

1950 Asymptotic properties) of the Wald«Wolfowits

tests of randomness

Annala of Mathematical Statistics 21 0 231-246

o

W„ Allen Wallis

1942 Compounding probabilities from independent

significance testso

Ee©no>m® trika 109 229=246

o
'

.

k a V

'

.

.... . s

-.i

'•

:

•

•

•
THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act

of Congress, March 3, 1901, as amended by Congress in Public Law 619, 1950.

These include the development and maintenance of the national standards of

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