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; .G7 W olfowitz, Jacob, 1910-
1942 Additive partition functions and a
; .V.'7 class of sta tistica l hypotheses...
cilew York3 1942.
62 ,da typewritten leaves. 29cm.
Thesis (Ph.D.) - New York university,
Graduate school, 1942.
"References": p .e63a
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Xerox University Microfilms, Ann Arbor, Michigan 48106
THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED.
ADDITIVE PARTITION FUNCTIONS AND A CLASS OF STATISTICAL HYPOTHESES
by / l Wolfov/itz
1942.
dissertation in the department of mathematics submitted to
a
the faculty of the Graduate School of Arts and Sciences in
partial fulfillm ent of the requirements for the degree of
Doctor of Philosophy.
ADDITIVE PARTITION FUNCTIONS AND A CLASS OF STATISTICAL HYPOTHESES
by J. Wolfowitz
1. Introduction
The purpose of the first part of this paper is to prove several
theorems about a class of functions of partitions which are additive
in structure and subject to ^ ld restrictions. These theorems may be
regarded as contributions to the theory of numbers, but if one makes
certain assignments of probabilities to the partitions the theorems
may be expressed as statements about asymptotic distributions. It is
in this latter, probabilistic language that we shall carry out the
proofs, for several reasons. The discussion vail be more concise and
certain circumlocutions w ill be avoided. The theorems have statistical
application and a number of theorems discussed recently in statistical
literature are corollaries of one of our theorems.
In the second part of this paper the theory of testing sta tis
tical hypotheses where the form of the distribution functions is totally
unknown and only continuity is assumed, w ill be discussed. The exact
extension of the likelihood ratio criterion to this case will be given.
Approximations to the application of this criterion in two problems will
be proposed, one of which applies the results mentioned above. Lastly,
, I
in connection with the second problem, a comjinatttpal problem w ill be
solved which is new and has interest per se.
2. Partitions of a single integer.
Let n be a positive integer and
■1
be any sequence of positive integers <t- ( I - 1, 1 , ..............., * )
i/ok.».ine
S
L -i
and s may be any integer from I VI . Two sequences A which have
different elements or the same elements arranged in different order are
ZH — I
sequences A. We shall consider the sequence A as a stochastic variable
and assign to all sequences A the same probability, which is therefore
^-*41
Let Yj be the number of elements CU in A
which equal j so ‘fclia‘fc "f* ^
is a stochastic variable. Let be an integer — y\ . Then the
joint distribution of the stochastic variables } Y \ ^ ^
is given as follows: The probability that ^ Jr. ( i s I/ Zf .. K j
iS
( M ' W ' - M
where the inner summation is carried out over all sets of non-negative
integers V V " ' t ~ S u c k ^ f
(Z't'j i t l K
+ + • ■ • + + >(K *.|)'t' ' ■ ' ' + r »u ^ T "
4
( J . i ) +
(The t - ' of course, are non-negative integers.)
- 2-
Let
1/u
v - Z . n
«, =1
and
(K-Kl)
^ ' c=K+l
so that Of and O p/ . . are both stochastic variables. The
(£ + 0
probability that at the same time
(2.fJ V\ *4 i G ' 1, ■ ■ ■ > K)
and
^ r ) V i j a Z k +1)
is given by f ^ . l ) •with the restrietion
M V-)+ " +^ * ^K+<)
added to the restrictions ( 2 . i j and C2. . 3 J
With this added restriction the summation in ( 2. . 1J may be performed
as follows: Let
t
, ± u L
L‘ 1 I
It is easy to see that the number of sequences A where every > k
iz
Y~ — T ^ v 0-vA ~ -w —
^ (K+j) (£+1) » c
is given by the coefficient of ^
in the purely formal expansion in )( OT
/ 1
f . *C+I <+>■ <+3 1) (i<+lK?K+|] ( \ \ (K+l)
( * + * + * + ...............) L ' ( ^ r )
3
and is
p - i
Hence p ““d ^ a 7 r ] 3 “
where this symbol w ill always denote the probability of the relation in
braces, is seen to be
(i-l)
& f '
If \ is a stochastic variable, let r~ I , A J and
\
cr I L V -
I denote, respectively the mean and variance of
(if they exist) and if is another stochastic variable, let
QT ( X ^ j ^ be the covariance between r and
Also let V tr -Xr- \ A / . By the
J f I
~
distribution function of we shall mean a function a-J L^y
such that = )
These conventions being established, we seek first to evaluate lH ^ I
This may be done by differentiating with respect to M the coeffic
ient of ^ in the purely formal expansion in v )C, of
2 7 * * 1 ^ X -h x ^~ + . - 1 ^ 1 4 - ‘j f ^ * ■ ' / * "
setting tj | and summing o^er all values of V
_4-
We have therefore to evaluate
^ r
- 'Vt-H .y - ^ 1a t — |
t
r = 5 L X
which is easily seen to give us the result
(l.fj E (V^J= (x-i- t a ) J- G<*)
while it is obvious that
/ I . — V_ +" I
( M j E" Cr »J = A
By use of similar devices the variances and covariances of the
r . may also be obtained. We omit the details of these
t
calculations and also the presentation of the covariances, since the
latter are not necessary for the proof of Theorem 2. The results ares
( 2 . U ) ^ ( ^ 3 - 2 - c
The lim itation on the value of L is necessary because the process
es for summing binomial coefficients with the aid of the device des
cribed above are no longer applicable. The matter is easily se tied,
however, for is a stochastic variable which can take only
the values 0 or | , then
-5-
< r ' ( X - ) ’ E ( i T C X j j
TheT fort>2iare such variables, so that
( i . i0 ^ ( y i ) = ■ ^ ' < - ■ ' - 3
a i+(
)
M Q " ~ ' - 1
\ 3 -a-x-i.
so -without difficulty we have^—— __
( w + t j
d.,,1 ^ Q - j ) -
-y\~ U
vihen is even and ^ 2. » an<^ c ^ X .
0LJ*i £ (v-J = *m ± i <rz(rl
( a . / j = ^
Finally,
V* / t \ r i —* “/
u . / t l E" (T(k+J = C x ~ ' / 2-
Tne next results we shall need mmaay be expressed in the follow-
mgr
Theorem 1, As 'J/V/approaches infinity, the joint distribution of the
stochastic variables . . .. . W , ) < K any fixed positive integer),
approaches the multivariate normal distribution.
This theorem is proved as follows: Make the substitutions
K J
% L = y- - -H-
\R
/ o - K - l
K
(KflJ,
in the expression
r- + /r>
V\4-l
—
11' ~ I
- /
which comes from and regard as equal to
L~l
-6-
J
p
Replace the various factorials by their asym^otic approximations
as given by Stirling’s formula and simplify the resulting expression.
The subsequent procedure is simple but laborious and we omit the de
ta ils, which are like those of the classical proof of De Moivre’s
theorem as given, for example, in Frechet Q ] , p. 89,
We now prove the following theorem an additive partition functions
Theorem 2, Let-|^(jhe a function defined for all positive
integral values of ^ which fu lfills the following conditionsr
a) There exists a pair of positive integers, a and b, such
that
b) the seriea^ , —<-
^
(j.ui
J/ll
converges. Let pT n » a function of the stochastic sequence A, be
defined as follows: p /■ \
(X'll) P ^C a) ~ ‘r r , 1
Then for any real the probability of the inequality
F 0 ) < I
approaches
1 ' ' JU
—co
as n
We restate this theorem without use of probabilistic terms:
Let A be any sequence of positive integers whose sum is a giVen
