UNBIASEDNESS OF TESTS OF HOwfOGENEITY WHEN ... - CiteSeerX

UNBIASEDNESS OF TESTS OF HOwfOGENEITY WHEN ALTERNATIVES ARE ORDERED

by

Arthur Cohen Michael D. Perlman Harold B. Sackrowitz

TECHNICAL REPORT No. 114 December 1987

Department of Statistics, GN..22 University of Washington Seattle, Washington 98195 USA

Unbiasedness of tests of homogeneity when alternatives are ordered

ty

Michael D. perlman 2 University of Washington

ion: Key words:

~~~~'dLch

2Research

Harold B. Sackrowitz Rutgers University

l

Primary'. 62F03

homogeneity, unbiasedness, similar test, Neyman structure, POlya frequency function of order two (PF z ) , log concave, multivariate total pos ivityof order two (MTP z ) , FKG inequality, admissibility, Bartholomew's test.

supported

DMS-8

8

6

by NSF Grant DMS-86-03489

1

Abstract

alternatives

r

ordered

Arthur Cohen, Michael D. Perlman, and Harold B. Sackrowitz

Xl'X 2""'Xk be independent random variables whose an exponential family with parameters

Let dens

8 l,8 2 , ••• ,8 k, f{x i I 8 i)

=

x.8. c(8 i)e

~

~

g{x i)·

82

H: 8 1

L ••• L Ok'

orderi for

j

is said to be ISO* if that if

~(x)

x »* y

by

is a Polya frequency

1

2

and define a partial

if and only if

and equality for

x »* y

K: 8

vs. the alternative

x = (x l,x 2 , ••• ,x k)

Write

1,2, ••. ,k-l

g

Consider testing the null

= 8 2 = ... = 8 k

on

=

Assume that

(PF ) . 2

function of order two hypothesis

That is, the densities are

respectively.

implies

j

=

k.

~(x)

A function

2

~(y).

is a similar test which is ISO* then

~(x)

We prove ~

is

The result contrasts with and complements the result of Robertson and Wright (1982).

They prove that when the density

nrl"\T'I,crty {normaL and ISO* power fi

st

ours,

r

a

ions. of

str

ions for from ours.

r r icat

the

, for

r

r

rti

r

ts

given.

Also some

particular distributions.

i For example it is proven that

Bartholomew's test is admissible for the normal case.

,X valued

2,

••• ,X

be independent continuous or integer-

k

distributed according to a one-parameter

var

exponential family with parameters

X.

the joint density of the

1.

9 i, i

=

tis,

l,2, ... ,k.

is

(1.1)

X

,XkJ,

1,X2

X.1.

mee.su r e for each

a = (9 1,9 2"., ,9 k).

The dominating

is Lebesgue measure on

for the

continuous case and counting measure on

{O,±l,±2, ... }

case where the

Assume that

X.

are integer-valued.

1.

Polya frequency function of order two (PF 2 ) i that is, log concave on

(-00,00)

or

{O,±l, •.. },

respectively.

for the g

is a g(~O)

is

The

rict Lnequa.Li

with at

unbiasedness and admissibility of tests. tson and Wright (1982) study the problem of test vs.

(i) the

ei ,

ions of the

(i L) the

of the

ion of

X

= a

(Xl"'"

)

is

1.

come from a sat ), or the

sson, distr

X.

H

tinomial. r

a

certain

. (x)

r

J ~

Y

tj{Y)

j

=

for

S.'U.~Qr

2, ... ,k.

as follows:

j = l,2, •.• ,k-l x »* y

tests.

More precisely, let

We define the par

The vector

is said to be 180* if

, cal

This monoton

ies unbiasedness of

r

X »*

ts.

x »* y

and

if and only if

tk(x) = tk(y).

implies

ordering

~

hex)

t.{x) J

A function

hey).

h

For the

distributions they study, Robertson and Wright (1982) prove that ~(x)

if

is a test function which is 180* then the power

function

test is 180*. results of

s paper (Theorem ;2.3) is that

in (I.}), if

for

which is similar and 180* then

~

~(x)

is

~

test function

is an unbiased test.

Whereas

such a result is not as strong as a result which yields an 180* (=

monotone) power function, the class of distributions for which

the result holds is substantially different from those studied by ight

class

normal family wi the Poisson family with means common sample size

.1)

distr

e.~ ,

common variance and means

e.~ r

the binomial family with

nand probabilit

ei ,

the gaJRffia family i

(whi

includes the chi-square distribution with two or more f

), and many others.

The binomial family is

ts

In fact,

a

3 we

case an 180* test

an I

t

r

In

tz {I

1

s

distr

in (1. H.

considered for the

In that study,

convex

em of

functions which were similar and Schur

shown to be unbiased.

method of proof used here

will be different (and simpler) than the method

there.

In addition to providing a sufficient condition for a test to be unbiased for H vs. K, we give a method of generating such tests in Theorem 2.5. Whereas the model of the paper is stated in terms of a single observation for each population it will be seen that all results

true when we have

x.1

observations from each

x.1

is replaced by the sample

from the ith population.

This follows as a special

population, provided that each mean

n

case of Theorem 2.6, a result on unbiasedness of tests for the important case of unequal numbers of observations from each population. Applications to particular tests and to particular distributions will be made in Section 3. wi th common shape parameter

(~

for a certain natural class

1),

For gamma distributions

unbiasedness is established s for equality of the scale

paramet For the statistical model of this paper one can easily derive a complete class of test procedures from the r Eaton (1970).

i

cate

a~lu~~~ible

of

likelihood

the normal case we prove

test

t

(1959) is

ss

i

tests for the ts are

as

and Poisson in

2.

Section 3

cate

conta

ications to specific distributions and

ss

2.

ity of tests is discussed in Section 4.

Unbiasedness of tests For

statistical model described near (1.1) we note first

any unbiased test for H vs. K must be similar and therefore k

must have Neyman structure with respect to Lehmann (1986), Theorem 2, p.144). ze

Hence any unbiased test of

have conditional size

a

(see

T == 2;.1.= IX'1.

(given

a

T = t).

It is

ate conditionally unbiased of size

r

for

are unbiased of s

a

Our plan will therefore

a.

be to show that the similar tests under study will be conditionally unbiased. We now discuss the notion of a multivariate totally positive distribution as done in Karlin and Rinott (1980). FKG inequa.1ity. 1"'11nr'1"'ion defined on where each

1.

V

and

~fk),

i.e.,

~l

==

f(x:) x

~2

is a totally ordered subset of

~.

where

~(k)

Let

A

x V Y

=

x A Y

=

x

This notion is be a x

~k'

satisfying

are the corresponding lattice operations on

n(

,

)

t

•

•

• t

n

A functien

f

with the property (2.1) is called multivariate

ther

=

~(k).

itive of order 2 (MTP 2) on

totally

for i

=

l, ••• ,k

e.

or

~

In this paper, = {O,fl, ••• }

i

for

l, ... ,k.

From Karlin and Rinott (1980) we note that if g(x) if

~(k)

are MTP 2 on

=

f(x)

g(Xi,X j )

MTP 2 on

then

where

9

f(x)g(x) is TP 2

is MTP 2 on ~i x

on

~j'

f (x)

and

~(k). then

Also, f

), hence products of such functions are MTP 2 on ?l:(k).

Now

= X~=lXi'

Tj

j = l,2, ... ,k,

T =

•.• ,T k),

(T I' ••• r T • The range of T is again ~ (k) k- l) while the range of T(k-l) is ~(k-l). Let f (t(k-l)lt )

a.ndT( k ...l)

9

denote the conditional density of

LEMMA 2.1:

Assume that

k"'l} I t ,

This

Under

H,

T(k-l)

(1.1) hoLds and

is MTP 2 on

k

PROOF:

is

the density of

given

90 -

T

on

~(k)

(2.2)

) r

c -

(9 , .•. ,9 0 ) € H. 0

~(k-l).

ies that

) €

k

C

> 0

is

8

marginal density of

Tk

at

tk

concave 'on fixed

t

But

g

is TP 2 on

log

hence for

is MTP 2 on

k,

r",(t(k-l)lt

Now let o E K

is positive.

u

00 E H.

and

= f

0

(t(k-l)lt )/f (t(k-l)lt) k 00 k

Also, let

Hk

denote the family of

component-wise nondecreasing functions on

LEMMA 2.2:

where

(From (2.2), note that the denominator does

a O•

not depend on

k)

II

t

For fixed

the ratio

k,

~(k).

ro(t(k-l)lt

k)

lies

in

Hk- l•

PROOF:

where

Proceeding as in the proof of Lemma 2.1 we find that

Ct

(a) >

0

whenever the marginal density of

k

is positive.

Since

0

=

(01,02, ••• ,Ok) E K,

TK

at

tk

the result is

immediate.

II

The well-known FKG inequali MTP 2

(2.3)

ity

f

on

and for an

for

(k)

s

E(

Karlin

nott (1

0).

~(x)

Let

PROOF:

!X(k-l).

Then

~

It suffices to show that TK

t(k-l)

where

x.

which is IS0* in

us.

given

be a similar size

=

t

Since

k•

for fixed

~

t

k•