ADDITIVE PARTITION FUNCTIONS AND A CLASS OF STATISTICAL HYPOTHESES

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ILD3907 ; .G7 W o lfo w itz , Jacob , 1 9 1 0 1942 A d d i t i v e p a r t i t i o n f u n c t i o n s and a ; .V.'7 c la s s o f s t a t is t ic a l h y p o th e s e s ... cilew Y o r k 3 1 9 4 2 . 6 2 , d a ty p ew r itte n l e a v e s . 29cm. T h e s i s ( P h . D . ) - New Y o r k u n i v e r s i t y , G raduate s c h o o l , 1 9 4 2 . " R e f e r e n c e s " : p . e6 3 a A 84750

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Xerox University Microfilms,

Ann Arbor, Michigan 48106

T H IS D IS S E R T A T IO N HAS BEEN M IC R O F IL M E D E X A C T L Y A S R E C E IV E D .

ADDITIVE PARTITION FUNCTIONS AND A CLASS OF STATISTICAL HYPOTHESES by / l

W o lfo v /itz 1942.

d i s s e r t a t i o n i n t h e d e p a r tm e n t o f m a t h e m a t ic s s u b m it t e d t o t h e f a c u l t y o f t h e G r a d u a te S c h o o l o f A r t s and S c i e n c e s i n p a r t i a l f u l f i l l m e n t o f the r e q u i r e m e n t s f o r t h e d e g r e e o f D octor o f P h ilo s o p h y .

a

ADDITIVE PARTITION FUNCTIONS AND A CLASS OF STATISTICAL HYPOTHESES by J . W olfow itz 1 . I n tr o d u c tio n The purpose o f th e f i r s t p a r t o f t h i s p ap er i s to prove s e v e r a l theorem s ab o u t a c l a s s o f f u n c tio n s o f p a r t i t i o n s which a r e a d d i tiv e i n s t r u c t u r e and s u b je c t t o ^ l d r e s t r i c t i o n s .

These theorem s may be

re g a rd e d a s c o n tr ib u tio n s t o th e th e o ry o f num bers, b u t i f one makes c e r t a i n assig n m e n ts o f p r o b a b i l i t i e s to th e p a r t i t i o n s th e theorem s may be e x p re s s e d a s s ta te m e n ts a b o u t asy m p to tic d i s t r i b u t i o n s .

I t is

i n t h i s l a t t e r , p r o b a b ilis tic lan g u ag e t h a t we s h a l l c a rry out th e p r o o f s , f o r s e v e r a l re a s o n s .

The d is c u s s io n v a i l be more co n cise and

c e r t a i n c irc u m lo c u tio n s w i l l be a v o id e d .

The theorem s have s t a t i s t i c a l

a p p l i c a t i o n and a number o f theorem s d is c u s s e d r e c e n tly i n s t a t i s t i c a l l i t e r a t u r e a r e c o r o l l a r i e s o f one o f our th e o re m s. I n th e second p a r t o f t h i s p ap er th e th e o ry o f t e s t i n g s t a t i s ­ t i c a l h y p o th e se s w here th e form o f th e d i s t r i b u t i o n fu n c tio n s i s t o t a l l y unknown and o n ly c o n tin u ity i s assum ed, w i l l be d is c u s s e d .

The e x a c t

e x te n s io n o f th e li k e lih o o d r a t i o c r i t e r i o n to t h i s case w i l l be g iv e n . A pproxim atio n s to th e a p p l ic a t io n o f t h i s c r i t e r i o n in two problem s w ill be p ro p o se d , one o f w hich a p p lie s th e r e s u l t s m entioned above. L a s t ly , ,I i n c o n n e c tio n w ith th e second problem , a c o m jin a tttp a l problem w i l l be s o lv e d w hich i s new and has i n t e r e s t p er s e . 2. P a r t i t i o n s o f a s in g le i n t e g e r . L e t n be a p o s i t i v e in te g e r and

■1

be any sequence of p o s i t i v e in te g e r s

< t-

( I

-

, ............... , * )

1, 1

i/ok.».ine S

L -i

and s may be any in te g e r from I

VI

. Two sequences A w hich have

d i f f e r e n t elem en ts o r th e same elem en ts a rra n g e d i n d i f f e r e n t o rd e r a re H —I sequences A.

Z

We s h a l l co n sid e r th e sequence A as a s to c h a s t ic v a r ia b le

and a s s ig n t o a l l sequences A th e same p r o b a b i l i t y , w hich i s th e r e f o r e ^ -* 4 1 L et Yj be th e number o f elem en ts CU in A which eq u al

j

so

i s a s to c h a s t ic v a r ia b le . L et

— y\

be an in te g e r

j o i n t d i s t r i b u t i o n o f th e s to c h a s t ic v a r ia b le s i s g iven as fo llo w s :

} ^

The p r o b a b i l i t y t h a t

‘fclia‘fc "f* ^ .

Then th e

Y \ Jr.

^ ^ ( i s

I/ Z f ..

Kj

iS

(

M

' W

w here th e in n e r summation i s c a r r i e d out o v er a l l s e t s o f n o n -n e g a tiv e in te g e r s

V

(Z't'j i t +

V

"

+ • ■• +

S u c k ^

' t ~

l K+

>( K * . | ) 't'

f

' ■' ' +

r »u

^

T "

4 ( J .i)

(The

+

t-'

o f c o u rs e , a re n o n -n e g a tiv e i n t e g e r s . ) -

2-

' -

M

Let

1/u

v -Z .n «, = 1

and

(K -K l) ^ ' so t h a t O f

c= K + l

O p/ . . a r e b o th s to c h a s t ic v a r ia b le s . (£ + 0 p r o b a b i l i t y t h a t a t th e same tim e

V\ *4 i

(2.fJ and ^

and

r )

V

i j

a

is g iv e n by f ^ . l )

M

The

G ' 1, ■■■>K) Z k +1) •with th e r e s t r i e t i o n

V-)+ " +^ * ^ K+V

( a f t e r rem oval o f p a r e n th e s e s ) i s . asy m p to -

vm tim e s 4*V th\e l1 ao os4t- +term o Pf

4" 4 «n ±l\

H ence, f o r a l l

( > 2 + */

**

A ^ 2-

"K^ s u f f i c i e n t l y l a r g e ,

(l«J

i

a

33-

)

^

and

m u l tip l ie s th e f i r s t by

(? -« J V and th e t h i r d term o f th e r i g h t member by

(a-W

^

J/ - -

’ )

A

A '

I t i s easy t o see t h a t f d r la r g e b u t f ix e d

and a l l "Vt g r e a te r th a n

a low er bound w hich i s a f u n c tio n o f f o n l y , th e e x p re s s io n ^ i s l e s s th a n th e e x p r e s s io n

»

- 2. "3

Hence, in view o f ^ 3 ' ^-'6

th e sum o f th e f i r s t and t h i r d term s o f th e r i g h t member o f (j^ * v

fo r

t —

•

i s n e g a tiv e .

th e second te rm o f th e r i g h t member o f from

to

I

*

Now c o n s id e r what happens to

0J.r)

U

when

goes

I t i s m u ltip lie d by

O a a ) ( > - / “• - 1 1 w hich, a ls o f o r la r g e b u t fix e d and a l l bound w hich i s a f u n c tio n o f J&.

»

l a r g e r th a n a low er

o n ly , i s e a s i l y seen t o be l e s s th a n

C o nsequently '

( j . n j

Cs-rJ

I t can be se e n w ith o u t d i f f i c u l t y t h a t such a p assag e o f

to th e n e x t h ig h e r in d e x i s always accom panied by m u l t i p l i c a t i o n by e x p re s s io n s s im i la r to

, and

(

3

f o r w hich s im i la r i n e q u a l i t i e s h o ld and t h a t co n seq u en tly c

( j ztj

0 £

tr* (j~lc

and f o r s im i la r re a s o n s

-34«


0 / V

e tc .

-o

J

I t i s c l e a r t h a t th e le n g th s and lo c a tio n o f th e in t e r v a ls des­

c rib e d a r e im m a te ria l, p ro v id ed o n ly t h a t th e y do n o t o v e rla p . th e d i s t r i b u t i o n s o f

X T an d

A lso

w ith in each i n t e r v a l a r e im m a te ria l,

p ro v id e d only t h a t th e y a r e c o n tin u o u s .

A ll t h a t m a tte r s f o r f in d in g

i s t h a t th e number and th e o rd e r o f th e d is ­ ju n c t

i n t e r v a l s s h a l l be th e same as th o s e o f th e ru n s in

( i . e . , i n t e r v a l s o f p o s iti v e p r o b a b i l i t y fo r w ith i n t e r v a l s o f p o s i t i v e p r o b a b il it y f o r t e r v a l s o f p o s itiv e p r o b a b i l i t y f o r ^

^

and f o r

p e c t iv e l y th e number o f ru n s o f th e elem en t o f th e elem ent

0

\J

,

)^~

must a l t e r n a t e

*

The number o f in ­

T

m ust equal r e s ­

&ud th e number o f nuns

| , and th e p r o b a b il it y o f th e f i r s t i n t e r v a l on th e

l e f t s h a l l be p o s i t i v e f o r ^

o r fo r T

-4 2 -

acc o rd in g a s th e f i r s t run

in

V

V

i s o f elem en ts

0

o r o f elem en ts

| , w ith th e same r e ­

l a t i o n o b ta in in g betw een th e l a s t i n t e r v a l on th e r i g h t and th e l a s t ru n i n

\/

) and th e p r o b a b i l i t y o f th e s e i n t e r v a l s .

L et

^

be th e so u g h t f o r p r o b a b i l i t y o f th e i n t e r v a l w hich co rresp o n d s to th e

* Tfc I —*

ru n o f elem en ts

0

and

O .

th e p r o b a b il it y o f th e i n -

t e r v a l w hich co rresp o n d s t o th e I — ru n o f elem en ts o rd e r t o o b ta in

\wl

{

.

In

, i t i s n e c e s s a ry t h a t th e elem en ts c o n s ti tu ti n g

each ru n s h a l l f a l l i n t o i t s c o rresp o n d in g i n t e r v a l .

Then c l e a r l y

by t h e m u ltin o m ia l theorem

C H P { V ; ( I * , A,

*

I t-

L w here

(/ 5

2.

\

and w h ere, when

i s f ix e d , th e p ro d u ct

%

w ith r e s p e c t to m e n t. to th e

J

i s ta k e n ov er a l l runs o f th e c o rresp o n d in g e l e ­

The r i g h t member o f v • I C

i s to be maximized w ith r e s p e c t

, s u b je c t o f co u rse to th e c o n s tr a i n ts

( p i Then i t may e a s i l y be - r e r if i e d t h a t th e maximum o ccu rs when

GrtJ

pr =

( J * /, p)

'^/i

J

F o r , a f t e r m u ltip ly in g by a c o n s ta n t and ta k in g th e lo g a rith m we i n ­ tro d u c e two Lagrange m u l t i p l i e r s

f/L /

43

'

and

JU, /

so t h a t th e max-

im iz in g

'p . • a r e g iv e n b y th e e q u a tio n s [ JT, 3 J and I u\ V J th o s e o b ta in e d by e q u a tin g to zero a l l th e p a r t i a l d e r iv a tiv e s o f

j

I 1

- 'J

The l a t t e r a r e th e r e f o r e

;

a fo r a l l

J

, whence

O ' . - J

(T4J

' fo llo w s .

I t i s e a sy to se e th a t t h e

extremum th u s o b ta in e d i s a maximum and a ls o an a b s o lu te maximum. s o u g h t-f o r s t a t i s t i c

-r(Jl

a f t e r th e r e s u l t s

i s th e n th e r i g h t member o f

have been i n s e r t e d .

I t may be s im p lif ie d

by rem oving a l l f a c t o r s w hich a r e fu n c tio n s o n ly o f 'Vb ( s in c e th e s e w i l l th e n be th e same f o r a l l

Cr , r )

4

The

1/ y )

and

~l/b.

r e c a l l i n g th a t

* l/

°

I t w i l l be co n v e n ie n t to ta k e th e lo g a rith m o f th e r e s u l t i n g e x p r e s s io n , so t h a t w ith \

, we had re v e rse d

• we woui^ have o b tain ed th e p e r-

th e r o le s o f th e

X 11 *

m u tation

. I t i s easy to see th a t any s t a t i s t i c , say

r> I

snd

^

J

J

* ff

used to t e s t th e n u ll h y p o th e sis, must be a fu n c tio n only o f

iro j=

w ith th e added p ro v iso t h a t

, „

(F ) Under th e fI - "Vj ^

(The rank c o r r e la tio n c o e f f ic ie n t is such a s t a t i s t i c ) . o

(

^

^ th e n

I

Q i+ i '+ i j

;

Cl t J if

i

i

>i

The ru n w i l l be c a lle d an asc e n d in g ru n or a descen d in g ru n a c c o rd ­ in g as

+

A ru n o f le n g th

|

^

^

i s o f e i t h e r ty p e , a t p le a s u r e .

£=

'/ ^ (e

The f i r s t ru n i s

, th e l a s t ^

2.

* > ry ^ is

3 ^2 - a d e scen d in g r u n o f le n g th

( a ru n o f le n g th o n e, a. , * Q > J/ , . ^ r h r . b tL T iOv^— ^ M y ^ J i s a de gene r a t ^ f u n c t i on such t h a t th e

r e l a t i o n betw een

X

i s f u n c tio n a l ( t h i s i s a s p e c ia l

case o f s to c h a s t ic r e l a t i o n s h i p ) » That i s where

F o r exam ple, l e t

% 3/

second ^

a n ascen d in g ru n o f le n g th two j ^ t h r e e , and

~~ 1 ___________ o * 4 - | .

t o say , X

*

f> ( X

f\ (— \ v ) which c o n ta in e x a c tly ^ L

ru n s. C o n sid e r, f o r 2 3 4 6 5 1 .

exam ple, fo r th e case ^

^

, th e sequence

Vie s h a ll say t h a t t h i s sequence c o n ta in s th e "con­

ta c ts " (2 ,3 ) , ( 3 , 4 ) , ( 6 , 5 ).

In g e n e r a l, a c o n ta c t i s d e fin e d as

th e ju x t a p o s i t i o n , in t h e sequence

o f c o n s e c u tiv e num bers, w hether I f i i . i s t h e number o f ru n s and

in a scen d in g or d escen d in g o r d e r .

th e number o f c o n ta c ts i n a sequence

th e n o b v io u sly

(1,|J L et ^ q be th e sequence

................. ''Yv--' o f th e f i r s t y\ \ y- i n -

1, 2,

te g e r s i n aso en d in g o r d e r .

The

''Yv.— |

c o n ta c ts o f t h i s sequence * of

R

c o n t a c ts , t h u s : (1 , 2 ), ( 2 , 3 ) , ................................................ me a n in g '^*1

(" h — | ,

TV

- iw............................................upp........ifn

of th e c o n ta c ts which c o n s ti tu te th e sequence

JV

) .i Suppose a r e s e le c te d

The re m a in in g

t a c t s form the com plementary s e t

* A fte r t h i s s e le c tio n th e s e ­

quence

| —

con-

in some manner t o form th e s e t Q .

R * may be c o n s id e re d a sequence of th e type o f th e se­

quences ^

o f S e c tio n 5 w ith th e members of Q

o f th e elem ents th e elem en ts I

w rite i t as

0

and t h e members o f

. Tflhen

R*€>J

p la y in g th e r o le

p la y in g th e r o l e of

is c o n sid ere d i n t h i s manner we w i l l

•

The d e f in itio n o f a run o f Section. 6"

56'

■

imehts W ts

th e Member

ana

\/N h e s j

/M .s e 5r a te -

£ irfo o tio a E> as a p p lie d to seq u en ces c a b le to

i s now a p p li­

We w i l l c a l l any su ch ru n o f th e members o f

o r of

(3

a group. We w ish f i r s t t o answ er th e fo llo w in g q u e s tio n :

O

b e s e le c te d from among th e

t h a t i t w i l l c o n ta in ^

A

I f , f o r a g iv e n d iv id e d in | ^ 0 j

-N

=

t

elem en ts o f b * [0 1

members a rra n g e d i n

•/

so

in C groups?

7

A '

, C be th e number o f groups i n t o which U

R * C o J , i t is c le a r th a t or

In how many

.

C— L

is

can e q u a l o n ly

Hence o n ly f o u r s i t u a t i o n s can a r i s e , as f o l ­

lo w s: a)

t /

. o f elem ents o f 0

th e r e f o r e composed y

elem en ts can be d iv id e d i n t o t

i s th e c o e f f i c i e n t o f

The f i r s t group in .

The number of ways i n w hich ru n s o f th e ty p e o f S e c tio n 2

in th e p u re ly fo rm al e x p an sio n o f

+ - -■ j and i s th e r e f o r e

rz-i \I \ .

-

(y -j

i

V t-I I elem en ts can be d iv id e d i n t o w ays.

R * ( 0 ) is

C

j?

. S im ila r ly 'V t *—I v, ■ f'+ .'l - ' O "f"' I ru n s i n I £

Hence t h i s s i t u a t i o n w i l l a r i s e in

jJ

^~J

ways.. * /

b)

L

occur m

“

C—I

•

By a s im ila r argum ent a s ab o v e, t h i s can

L_ l -

-67-

=;

c*

^

ti* jv**-

•e j&.

k

^ r o

. T k r w*

0

_y/which con­ ta in

e x a c tly ^

c o n t a c ts .

As was s a id b e f o r e , th e t o t a l of th e num­

b e r of sequences in ea c h is

•

-5 9 -

L et

"^e s e ^

^e

sequences i n a l l th e s e f a m i l i e s , w ith eac h sequence i n

o u n t-

ed as many tim es a s th e number o f f a m ilie s i n w hich i t o c c u rs . e r y sequence i n

has th e

g e n e ra te d i t , b u t a f t e r p erm u tin g Hence e v e ry sequence in by C M

m

( 3 w hich

o th e r c o n ta c ts may s t i l l e x i s t .

h a s a t l e a s t ^ c o n ta c ts and th e r e f o r e

, a t m ost

has e x a c tly

c o n ta c ts o f th e s e t

ru n s .

C le a r ly , a sequence w hich

c o n ta c ts o c c u rs e x a c tly once i n

, s in c e i t

can a p p e a r o n ly i k th e fa m ily g e n e ra te d by th e s e t t a c t s and i n no o th e r fa m ily .

Q

of its

tim e s in

f o r i t w i l l ap p ea r once i n each fa m ily g e n e ra te d by a s e t c o n s is ts o f one o f t h e ^ ^ M

s e le c tio n s o f

(^, ■+■2^ J

fitAtim e s ,

and so f o r t h .

Q

which

c o n ta c ts from among

c o n ta c ts , and in no o th e r fa m ily .

quence w hich has e x a c tly

con­

A sequence w hich has e x a c tly

c o n ta c ts w i l l ap p ea r e x a c tly

its ( e

Ev­

S im ila r ly each s e ­

c o n ta c ts w i l l ap p ear i n

«(«/

We t h e r e f o r e o b ta in , i n view o f C M ,

i

C 2/ The system o f

l i n e a r e q u a tio n

d eterm in es th e q u a n titie s

N o | ,