不等式制約を持つ正規分布の平均の統計的推測に関する研究

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九州大学学術情報リポジトリ

Kyushu University Institutional Repository

不等式制約を持つ正規分布の平均の統計的推測に関する研究

岩佐, 学

https://doi.org/10.11501/3111017

出版情報：Kyushu University, 1995, 博士（数理学）, 論文博士バージョン：

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STUDIES ON STATISTIC AL INFERENCE OF

NORM AL ME ANS WITH INEQUALITY CONSTR A INTS

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Studies on Statistical Inference of

Nor mal Means with Inequality Constraints

Manabu IV/ASA

Department of Mathematical Science Faculty of Engineering Science

Osaka University

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Preface

vVhen the space of parameters to be inferred is restricted by inequality constraint uch as

order relations, it is expected that more effective inferences are possible by using the information.

Statistical theories have been developed to explore this possibility during the last forty years and the inference under inequality constraints is recognized widely as an important. field of mathematical statistics (cf. Barlow, Bartholomew, Bremner and Brunk (1972) and Robertson.

vv'right and Dykstra (1988)).

In this thesis, monotonicity properties of the power functions of likelihood ratio tests (LRTs) and admissibility of a maximum likelihood estimator (!viLE) for normal means vvith inequality constraints are consiJered.

Suppose a k-variate normal population

Nk

(J.L, 0"2

Ik).

The study of the LRT for ^J.L⁼ 0 against J.L ^EC, where C is a closed convex cone, started from Bartholomew (1959) and Kudo (1963) and have been dev(·loped extensively by a number of researchers. It is known that null distributions of the LRT star.istics are weighted sums of chi squared distributions when 0"2 is known and of Beta distributions when 0"2 is unknown. The LRTs based on the statistics are called x2-test and E2 -test, respectively. Although it has been verified by numerical investigations that the LRTs have tolerable performance in the powers, their distributions under alternatives are generally very complex and theoretical studies on their powers have not been carried out sufficiently. In the cone-restricted testing problem, it is very important to understand the difference of the power with respect to the direction of alternatives from the origin. It is conjectured that th<� power of the LRT is relatively large at an inner direction and relatively small at an outer direction in the cone C ( cf. Bartholomew(1961)). This conjecture has not been solved except for some special cases. On the other hand, the isotonic regression which is a method to derive i\ILEs under order restrictions has played a central role in the theory of estimation under inequality constraints.

Regarding the nptint<dity, it is speculated that MLEs under inequality constraints ar(' not very good. In fact, it is kuown that iviLEs are inadmissible for squared error los function. However.

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this is one aspect of the optimality of the estimator. Nothing is known about their admissibilit�·

when the loss functions are other than the squared error loss.

In Chapter 1, we develop probability inequalities which play the key roles in investigating the monotonicity of the po·wer functions of the x2-test and the E2 -test. Fork-variate normal random vector X ^� Nk

(

/1, I�,:) and a set A C R k, we consider the probability Pr{X E

.-1}

(denoted b�·

P(�-t;

A)).

In studying inequalities about P(�-t;

A),

it is essential to specify a geometrical propert�·

of A and an orderin.� introduced in the parameter space. For our purpose, it is required that the set A is asymmetric and that the ordering of 11 detects the change of 8 ⁼

11/

� as well as that of� = Jli!il,. The conjecture stated above is relevant to the monotonicit�· with respect to 8. In this thesis, we present two inequalities. One is useful to detect the both changes and the other is for detecting of the change of 8. A concept (called light-tailed property), which is obtained by removing the symmetry condition from G-monotone property introduced by Eaton and Perlman

(1977),

plays an important role in our argument. The first inequality is given ^a.

an assertion that if A. is light-tailed for ^a

cone K, P(J-L; A) is light-tailed

for K. The

second

inequality is gi\·en under a weaker condition of A than the light-tailed property. \Ye consider an ordering � K on a sphere induced by a cone K and prove that if A is increasing in � f\., P(J-L; A.) is increasing in � K. :Yloreover, we show that a spherical analogy of majorization ordering is defined as a special case of our ordering.

In Chapter 2, ^vvediscuss the monotonicity of the power functions (If the x2-test and the £2- test. (V/e also consider a case where the null hypothesis is restricted by a convex cone.) First, we present general results obtained from the inequalities of the previous chapter. Our results indicate similarities of the monotonicity property of the power functions of the x2-test and the E2 -test. Next, we consider a case where the cone C is sy mmetric with respect to reflections.

vVe define concepts of unimodality on spheres and show that our monotonicity results imply the unimoclality of the power functions when C is symmetric. This result on the unimodality is useful for discussing Bartholomew's conjectures described above. Some of the ^conj^ectures ^are settled affirmatively.

In Chapter :J, we study admissibility of a 1LE. Let X be a normal random variable JV

(p,, 1).

Under the conditiou that f.L belongs to a closed interval

[

^-m,

m],

the l'vlLE is given as the projection of ""\ onto

[

^-rn,

rn].

It is known that this l'vlLE is inadmissible and i1nproved by a shrinkage estimator under squared error loss function. vVe discuss admissibility of the l\ILE for

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loss functions other than the squared error loss. First, we give a sufficient condition for the 1\ILE to be improved by a shrinkage estimator. Then, for loss function

ep(X.JL)

=

lx- 1'·1'\

\Ve show that the .NILE is inadmissible and improved by a shrinkage estimator when p > 1, howe,·er, that the .NILE is a Bayes estimator and therefore admissible when p = 1.

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Acknow ledgernents

I would like to exrress my gratitude to Professor Takashi Yanagawa for his guidance, valuable suggestion and constant encouragement. I am also grateful to Professor Akio Kudo and Professor Keiiti Isii for their encouragement and kindness.

I have benefited greatly from the other teachers and colleagues too numerous to mention here. Finally, I wish to express my hearty thanks to all of them.

February 1996 i\Ianabu I was a

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Basic notation

All vectors ;.�.re in Boldface type and, unless specified otherwise, all vectors are column vectors.

x'

donates the transposition of the vector

x.

•

R=(-oo,oo).

•

Rk ={xI x = (x1,

· · · ,xk

)

', -oo

<Xi<

oo fori= 1, _{· · ·}

,

k

}

^.

•

N(J.1,

CT2

)

denotes a univariate normal distribution with mean �L and variance 0'2.

• JVk

(

J.L,

2:)

denotes a k-variate normal distribution with mean vector J.L and co,·ariance ma

trix 2:.

•

I

=

^{1, 2}· · ·

. n} . i=l

n

•

..C(al, a2,

· · ·

,

ar

) ={xI x = L ^tiai

^for^some

^ti

^E

^R,

ⁱ

⁼

^{1, 2,}^{· · ·}

^,

ⁿ

^}.

i=l

•

rd(x)

is the retlection of

x

through the hyperplane

..C(d)_i.

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Chapter 1 Inequalities on the probability content of certain asyn1n1etric regions for

norn1al distributions

1.1 Introduction

Theories of probability inequalities play fundamental roles in mathematical statistics and a wide variety of inequalities have been proposed with development of several important concepts, including convexity, symmetry, unimodality and ordering. See Tong

(1980),

Dharmadhikari and Joag-dev

(1988)

and Pecaric, Proschan and Tong

(1992)

for detailed reviews.

Suppose that X is ^ak-variate normal random variable

Nk(J.L, Ik)·

Our study concerns the probability Pr

{

^X^E

A}

for a set A of R

k.

Since the probability is regarded as a function of

J.L,

we

shall denote the prol>ability by

P(J.L; A).

In studying inequalities about

P(J.L; .-l),

_itis essential to specify a geometrical property of the set

A

and an ordering of the parameter

J.L.

(In general studies of probability inequalities, the distribution of a random variable X is also an important element. In this section, however, we consider only the normal case for simplicity. \\·e note that the results presented below were obtained under more general conditions of distributions.)

A pioneering work in this area is found in Anderson

(1955).

_Heproved that

(1.1)

_{for all}

J.L

_E_R

k

_and

0

_�s � t

if A is a centrally symmetric, convex set. Mudholkar

(1966)

extended Anderson's inequality by replacing central sytnmetry by invariance under a transformation group.

A further d1:velupment is done by weakening the convexity assumption of A.. iVIarshall and Olkin

(1974)

developed an argument based on majorization and Schur convexity. For

x =

(

x l,

· · ·, Xk)

_and_y=

(Yl, · · ·, Yk)

_ER

k,

x is said to be majorized by y, clenotc·d by ^x-<m. y,

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if k k L X[i] ⁼ L Y[i]

i=l i=l

and

n n

L X[i] :S LY[i] for all ^{n =}1, · · ·. k-1.

i=l i=l

where X[1] 2: · · · 2: X[k] and Y[l] 2: · · · 2: Y[k] are ordered components of

(

^.rl· · · · , .ck) and

(y1, · · ·, Yk)· Furthermore, a set A C Rk is said to be Schur convex if x -<m y and y

E

A imply x

E

A

.

.;\'Iarshall and Olkin (1974) proved that if A is Schur convex,

P(J-L: A)

is a Schur concave function, that is

(1.2) for all

J-L1

-<m

J-L2.

The ordering -<m is related to a symmetric group and their argument is generalized for ^are

flection group by Eaton and Perlman (1977). It is noteworthy that the mojorization ordering i�

multidimensional while the ordering used in Anderson's inequality is one dimensional.

In the inequalities (1.1) and (1.2), the symmetry of the set A. plays essential rolf's. However, considering applications for restricted testing problems, the symmetry conditions ^can not be assumed generally. Therefore we have to develop inequalities for an asymmetric et A.

The simplest argument for asymmetric

A

is based on a property of A such that

(1.3) A+

CocA

for some convex cone

Co.

If A satisfies (1.:3), iL is easily proved that

(1.4)

for all

J-Lo E Co.

This inequality is useful for deriving monotonicity results on the power functions of likelihood ratio tests in cone-restricted testing problems. See Section 2.6 of Robertson et al. (1988).

On the other hand, Iukerjee, Robertson and vVright (1986) gave another inequality. They considered a convex set

A

satisfying that for some d

E

R k

x- 21r(xl£(d))

E A

for all x

E An H!,

where 7r(xl£(d�·) is the projection ofx onto a linear space L:(d) ⁼ {td

ItER}

and

H!

⁼^{x

^E

R k

I

x'd 2: 0

}

. They proved that if a convex set

A

satisfies the above condition,

P

(

s

d

; A)

2: P

t

( d^;

A)

for all 0 :::; ^s:::; t.

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(Similar argum�nts can be found in previous works by Pincus (1975) ancl );omakuchi (198-1).) Recently, Hu and vYright (1994) extended the inequality of ;vrukerjee et al. ( 19 6) h�- \Yeakening the convexity assumption of

.A..

They proved that if

A

satisfies that for ·ome x0. d

^ER k

( 1.5)

it holds that (1.6)

x- t1r(x- xo I L.:(d))

^E.A.

for all

0 ::;

t

::; 2,

x

^EAn

(xo

⁺

H;j).

P(xo

⁺

sd; A.)

2:

P(xo

+

td; A) for all

0

:S

s :S

t.

In the study of the power functions of likelihood ratio tests in cone-restricted tc�ting prob

lems, it is very important to understand the difference of the power -vvith respect to the direc

tion of alternative. Although the above two inequalities are powerful tools to establish certain monotonicity properties of the power functions including unbiasedness, they are not useful for detecting such directional differences of the power functions.

In this thesis, we develop two inequalities which are useful for the stud:.· of directional

differences of the power functions.

In Section

1.2, "''e

show that the inequality (1.6) can be extended regarding ordering of

^J.L.

The inequality (1.6) has been regarded as an extension of Anderson's result (1.1) (cf . :\Iukerjee, Robertson and \Vrigbt (1986) and Hu and Wright (1994)). Our argument begins with regarding

a

set satisfying (1.5) as a generalization of G-monotone set introduced by Eaton and Perlman (1977). The generalized property, called light-tailed property, is defined b:v removing the sym

metry condition of G-monotonicity. A justification of our approach appears as an improvement of the ordering used in the inequality. Observe the difference of the ordering of (1.1) and that of (1.2). Our extenci ordering is multidimensional and detects directional differences of

^J.L.

In Section 1.3, we give another inequality. vVe define an ordering :SK ort a sphere indnced by a closed cone

]{

and prove that if

A

is monotone in the ordering :S K, the probability P(

^J.L;

.A.) is monotone in ::; K as a function of

^J.L.

Although the ordering :S K detects only directional dif

ferences, the coadition of

A

is weaker than that in the inequality of the previons s�ction. In

Section 1.4, by c:ombining results in the previous two sections, we show that if.--\. is light-tailed

for a cone

J(, J> ^·^J.L;A)

is light-tailed for

J(.

We clarify the relation between the inequality based

on the light-tailed property and the inequality (1.4). In Section 1.5, we

define

n.n ordering on

spheres analogous tc, majorization. vVe show that the spherical rnajorization ordering is realized

as a special case of

the

ordering introduced in Section 1.3.

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The probability

P(J_t: A)

is represPnted as a convolution of the densit.v function n(x) of

�V.L.:(O, h)

and the identity fllnction f..�(x) of the set

A,

i.e.

P ( ^J.L:

A ) =

^!.4^{* n}( ^J.L) =

./R

^kⁿ⁽^J.L^-^x

⁾

^£.4⁽^x⁾^rl^x.

Act ua.lly, some of the above i neq uali ties can be rest a. ted as properties that certain classes of functions are closed under convolution ( cf. Proposition

1.1).

The form of convolution is simple a.nd convenient. In this chapter, our fundamental results are derived as inequalities of the convolution. Throughout this section, we suppose that functions .f.g are measurable and the convolution f * g is finitely determined.

1.2 An extension of majorization inequality

Let rd(x) be the reflection of x through the hyperplane

.C(d)..L,

^{that is}

rct(x)

=

^x-27r(xl£(d)) =

�

where

JJdJJ2 = d'd.

(

^dd'

)

I-2lldJI2

^X

ifd�O,

X

ifd=O,

A transformation group acting on Rk generated by reflections is called a reflection group.

For a. reflection group G, we consider a cone A·c defined a.s

�ote that I1

.

^."c⁼-A·c and that A"c is not necessarily convex. :Vloreover. let \G = h"c n S1, where Sr

=

^{x^{E Rk}^I

llxll = r},

and Lc denote the smallest linear space including l\·G· For x E Rk, we denote by Convc(x) the convex hull of {T(x) IT E

C}.

The following definition and proposition are dne to Eaton and Perlman (

1977).

Definition 1.1 Let G UP n reflection group. For x. y E R .1.:, x is

said to

^{UP G'}-majori::Pd by y.

dPnoted

^byx ^�Gy.

if

x E ConvG(y). !v!01·e01 e1·. n

f7Lnction .f

^is

said to

^bP

G-monotone if

x �G y implir:s f(x) � f(y)'.

\NhPn (r' is a. sym tnPt ric group. :r �c y is eq ui valent to .r �m y a.s sta.tPd in t hP in trod nction.

T h 0 s t r a c t u r e of ('on v n ( x ) is studied i n S P r t ion 4 of £ aJ on n.n d Perl m a n

( 19 7 7 ) .

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Proposition 1.1 (Eaton and Perlman (1977)) If f and g are G-monotoTie, f

*

g(x)

=

}R r 1.: f(y)g(x- y)dy

is G-monotone.

If

f

is G-monotone,

f

is symmetric with respect to reflections belonging to G. v\·e consider an extension of Propc)sition

1.1

by weakening the symmetry condition.

FordE Rk, {ro,

r

d }

is a reflection group. Hereafter, we denote the reflection group

{ro, r d }

by

Gd.

For G

d ,

Ked =Led =

L:(d)

and

C

o

nv^ed

(x)

=

{tx

+

(1- t)rd(x) I

0 �

t

�

1}

⁼

{x- t1r(x I L:(d)) I

0 � ^t�

2}.

To begin with, WP-introduce a concept in which the symmetry condition of G-monot.oicity is removed. The condition

(1.5)

is described systematically by using the terms of G-majorization.

Throughout this chapter, unless specified otherwise, K denotes a closed and not necessarily convex cone in

R k.

Definition 1.2

A

fu.nction f is said to be light-tailed for

_K

if

f(x) � f(y) for all d

E K,

x

E

H;i and y

_-<ed

x.

Moreover, a set

_A.

is said to be light-tailed forK if the indicator function of

_A.

is light-tailed for

K.

A function

f

is G-monotone if and only iff is light-tailed for f(e. vVe note that the light

tailed property of

f

for J( does not imply the light-tailed property of - f for -J{.

For a function

f

and

d

E

R

k, define

{ ^f(x)

fd(x)

=

f(rd(x))

if X E

H;i

if X E

H;;.

We denote by C

(a1,

· · · ,

an)

the convex cone consisting of the points represented by nonnegative combinations of

a1,

·

·,an.

Proofs of the next two lemmas are routine and omitted.

Lemma 1.1 f is light-tailed for

C

(d) if and only if (1) !d is Gd -monotune and

(2) f - fd

_{2: 0.}

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Lemma

1.2

If f(xJ

is light-tailed for

C(d)

and

xo

E

H;j, f(x

+

xo)

is light-taded

for C(d).

Lemma

1.3

Ij

f is light-tailed for

C(d)

by Lemma 1.2. Since f = !d

⁺

(J- !d), we obtain

f * g(td) = !d * g(td)

⁺

(J- !d)* g(td).

From Proposition

1.1

and Lemma

1.1,

!d * g is Gd-monotone, and therefore fd * g(td) is nonin

Definition

1.3 For

x, y

E Rk, define ^X

-j_[( y

i

f y- X

E K n

-j_K

is not transitive even if J( is convex. This suggests t

hat. the

convo

lution inequality holds under finer binary relations.

Definition

1.4 Fo1·

x,y

^E ^Rk and a cone K C Rk, define

x

^<<g

y

if there e.r-ist

⁼

lim Xn 1

and

Xp = lim Xn

k .

n---cx '

n-+oo

^'ⁿ

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Adding the continuity assumption to f ^*g, we immediately obtain a rc� nlt.

Proposition

1.2 Su.ppose f is light-tailed for

f{

and g is for±!\. Jf f ^*

g

is cont?:n7W'll.s, for

x

_<<Ky.

Remark (1)

The c()ntinuity condition of f ^*g is satisfied if one is int�grable and the other is bounded.

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^If^gis light-tailed for ±K, g is G-monotone for the reflection group G generated

by { rd I d

E K}. Therefore, when K has a positive k-dimensional volume,

g

is light-tailed for ±!( if and only if g is spherically symmetric and unimodal, i.e.

g(x)

=

go(llxll)

for some nonincresing function

go

on

[0,

_oo).See Theorm

3.1

of Eaton and Perlman

(1977).

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For a reflection group G, it is known that

x

-<c

y

_implies

_x

<<Kc

rct(Y)

for some

rd

E G (cf.

Eaton and Perlman

(1977,

Section

4)

and Marshall and Olkin

(1979,

Section

14.C)).

In the follovving, we shall give a result which indicates that <<K induces an order relation between two points on Sr. Note that -jK does not have such a property.

Suppose that J( is a two-dimensional convex cone

C (a1, a2),

where

a1

and

a2

^arelinearly independent. Then we define

f(i ={xI a�x

2: O,

a �

_{_}_i

_x

_:::;

_0}

_fori=_1,2

First we note that the ordering <<c(a1,a2) is essentially an ordering on the two-dimensional affine spaces

x

+

.C(a1, a2) (x

E

Rk).

Accordingly, we have to study the ordering on a one-dimensional sphere

(x

+

.C(al, a2))

n sllxll· ^Since^X << K

y

_and^X_-jKyare equivalent if ^X

¢:.

I\, the problem is the case

x

E [(.

For

x

E Ki, de fin�

Px,K

=Kin

Sllxll n

(x

+

f().

The set Px,K is a s11 bset of an arc Ki

n

Sllxll n

(x

+

.C(a1, a2))

consisting of

x

and the points nearer to the center of J( than

x.

Lemma

1.4 If K ⁼

C(a1, a2),

where

a1

and

a2

are linearly independent, and

x

E J(i, then Px,K ^C

{y I

X << K ^y^}.

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Proof Suppose:· Px1 ^,\.-=/:-

{ x}

and fix any y

(

-=/:-

x) E

PxJ\ .. Let

D = xo

⁺C

( x - xo,

y

- xo).

Then

xo #- x

and

D =Kin (xo

⁺

£(a1,a2)).

For each

n,

we define a finite sequence

{xn 1j};·�1

as follows. First we define

Xn1l = x ( E D \ { xo}),

Xn1j+l = Xn1j

₊ⁿ

-1d n1j, j =

_1,^·^·^·

,kn-

1,

where

dnlj

be the point of J(

n £(xnlj)

^l..

ns 1

and

kn

is the smallest number such that the segment

(xn1kn-1' Xn1knl

intersects the half line

xo

⁺C

(

y-

xo).

Note that J( ⁿ

L(xn1j)l..

is ^ahalf-line if

Xn1j

^E_Ki. ?vioreover, for

Xn1j,

we define

Pn1j E [0, 1r]

^by

(xn1j - xo,

y-

xo)

COS

Pn j =

^.

and then

kn = O(n)

as

n

^{---7 oo.}

Next we show that the sequence

{ {xn1j};�d�=1

satisfies the conditions of Definition 1.4. It is obvious from the definition that

Xn1j

_�^K

Xn1j+1

and

n--.oo

lim

Xn 1

¹

= x.

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l'vioreover, since

_kn =

O(n) and

we have

This implies that liu1 X

n k ⁼

y. Therefore, we obtain x

<<I<

y.

I

n-c'() ^, ⁿ

Lemma

_1.4

implit>s that for any x

E Rk,

the arc I\insllxlln(x+L:(a1,a2)) is totally ordered by

<<I<.

1.3 A cone ordering on spheres and a related inequality

In the previous section, we saw that the ordering

_<<K

induces order relations on

_Sr.

In this section, we consider another ordering on

_{S r}

induced by a closed cone K.

Definition 1.5

We define x

_�K

y if llxll

⁼

IIYII and there exist finite x1,

^{· ·}^·^,

Xp

^ESllx/1

such that

Xt

⁼

x, Xp

⁼

y and Xi+l -Xi

^EJ(

for all

i ⁼ 1, ^{· ·}^·, ^p- 1.

vVhen, for example,

_J(

is a finite union of convex cones, the ordering

_�K

on

_Sr

is finer than the restriction of

<<1,·

on

_Sr.

vVe note that th(· convexity of K is not assumed in the above defillition and that x

� ^1\·

y is not necessarily equivalent to x

�K

y where

_J(

is the convex hull of K. Obviously x

�I<

y implies tx

�I<

ty for all

t >

0. In this sense,

_�K

can be regarded as

au

ordering of direction from the origin.

Let J(*

⁼

{x

E Rk

I x'y

2:

0 for ally

E

K}.

Lemma 1.5 K*

is identical with the set consisting of all maximal points with respect to

� ^f(,

z.e.

K* ⁼

{

^x^ER k

I x

� K

y implies x

⁼

y}.

Proof

We denote rhe right hand of the equation by U(K). For x

^EIC',

(x+J()nSIIxll

⁼

{x}.

Therefore,

_J(*C

U(K). On the other hand, when x �

_J{*,

there exists y( # 0)

E

J( such

_that

x'y

<

0. Then x

_��'

x + ty for positive

t ⁼

-IIYII2 /(2x'y). This implies that x � U(I\). Hence

U(I<)

c

K*.

^I

(19)

Definition 1.6 A

fu.nction f(x) is said to be increasing (decreasing) 1:n �K if f(x) � (?_} f(y) for all x �f( y .

Moreover,

a

set

�K.

Fix any d ^E f{ _andx E

H;;.

^From

the assumption�: of j and

g,

we have

f(y)

?.

f(rd(Y)) g(y)

=

g(rd(Y))

for ally ^E

H"d,

for ally ^ERk.

Moreover, since

x- rd(Y)

^EConvcd

(x- y)

for

x

^E

H;;

^andy^E

H"d,

g(x- y) � g(x- rd(Y))

^{for ally}^E

H"d.

From the above relations,

f

*

g(x)- f

*

g(rd(x))

r f(y)g(x- y)dy- r f(y)g(rd(x) - y)dy

iRk iRk

.IRk f(y)g(x- y)dy- JRk f(y)g(rd(rd(x))- rd(Y))dy

r f(y)g(x- y)dy- r f(rd(y))g(x- y)dy

iRk iRk

(20)

./Rk {f(y)- f(rct(Y))}g(x- y)dy

{,

⁺

{f(y)- f(rct(Y))}g(x- y)dy +;;

^_

{f(y)- f(rct(Y))}g(x- y)rly

-� �

{ {f(y)- f(rct(Y))}g(x- y)dy

+

{ {f(rct(Y))- f(y)}g(x- rct(Y))dy

J H: JH:

{

+

{f(y)- f(rct(Y))}{g(x- y)- g(x- rct(Y))}dy :S

0 .

.!

f!d

Therefore, the result. follows from Lemma 1.5. I

Theorem 1.1

Suppose

X

is a k-dimensional normal random variable JV(!-L,

0"2

I)

and A _CR _k

is increasing ( decreasinq) in :SK. Then

Pr

{

X E

A} is increasing (decreasing) in

:::; _g

as a function of

1-L·

1.4 A further extension and a comparison between inequalities

If a function

f

is light-tailed for

K, f

is decreasing in :SK. Therefore, combining Lemma 1.3 and Proposition 1.3, we obtain the following assertion for light-tailed functions.

Proposition 1.4

Iff is light-tailed forK and g is for ±K, f * g is light-tailed for I\.

Proof It is sufficient to show that for any d E

K

_and

x

_E

H"d,

f * g(x) :S f * g(y)

for

ally

E Convcd

(x).

When

y

_E

Hj·: ^f

^:+

g(x) :S f * g(y)

from Lemma 1.3. On the other hand, when

y

_E

H;;,

f * g(rct(Y)) :S f * g(y)

from Proposition 1.3. _Since

rct(Y)

E

H"d

and

rct(Y)

E C_onvcd

(x)

, we obtain

f * g(x) :Sf* g(rct(Y)) :Sf* g(y)

I

Theorem 1.2

Suppose

_X

is a k-dimensional normal random variable JV(/-L,

0"2

I)

anrl A. C R k

is light-tailed for

J(.

Then

Pr

{

X E

A} is light-tailed forK as a function of

1-L·

In the following, we shall compare Theorem 1.2 with the inequality

(1.4).

For a closed cone J(, we define

X <K

y

_¢:==;>

y

-X E

K.

(21)

:S

IIYII-

In the following sense, the light-tailed property is a generalization of the increasing property.

Lemma

1.8 The following are equivalent.

(1)

f(x)

is increasinq forK.

(2)

f(x + xo)

^?·^•^.,light-tailed for -J( ^{for all}

xo E ^Rk.

(3) - f(x + xo)

is light-tailed forK for all

Xo E ^{R k.}

Proof

Since (1) is equivalent to that ^-

f(x)

is increasing for -K, the proof is complete by showing the equivalence of

(1)

and (2).

[(1) =:;. (2)] \Ye can assume

xo =

0 ^because

f(x + x0)

is increasing for J{ ^forany

xo E ^Rk.

^Fix

dE

^{K and}

x E H:d.

^Sincef is increasing for

C(d)

^and

1r(xj..C(-d)) = -yd

^forsome ^'Y :S 0,

f(x- t1r(xj..C(d)))

2:

f(x)

^{for all}0 :S

t

:S ^{2. Hence}

f(x)

is light-tailed for -I\.

is not light-tailed for

-C(d).

^I

For a fixed. conn�x J(, we see from Lemma 1.7 that the ordering used in (1.4) is finer thctn that in Theorem

1.2.

However, Lemma 1.8 implies that the light-tailed propert.y holds for a

wider cone (not nece�sarily convex) than the increasing propety does. Therefore, onr inequality

(22)

produces many resulrs which are not obtained from thf' inequality

(1.4)

_Forexampk. some of the results in Sections 2.3 and 2.4 are obtainable from Theorem 1.2 but not from thf' irwqualit�- (1.4).

1.5 Majorization on spheres

G-majorization ordering

-<c

defined in Section 1.2 gives an ordering on a linear space

Lc

^and

its translations. In this section, as an analogy of the G-majorization, we define a.n ordering on a sphere induced by a. reflection group

G.

For

G, J(c = {

_d_E_R_k

_I

_rd _E

_G}, _{Ac =} {

_d_E

_{51 I}

_rd _E

_G}

_and

_Lc

_{is the}_smallest_linear

space including

Lc.

\IIoreover, we denote by

.A(G)

the dimension of

Lc.

In the follov\-ing, we consider the case where

.A( G)

:S k - 1 and suppose that

g

belongs to

S1

n

L�.

Definition 1.7

For x ,y

^E

Sr, xis said to be (G,g)-majorized by y, denoted by x -<(c,g) y, if x

Ht and -rr(xiLc) -<c

^z

-<c -rr(yiLc) for some

z E

Lc.

Transitivity of

-<rc,g)

is obvious from that of

-<c.

Lemma 1.9

x -<(c,K) y is equivalent to that there exist finite xo(

⁼

x), x1, · · ·, xp( = y) such that

Proof vVe considc�r the case (1). The other cases are proved similarly. From Lemma 4.5 of Eaton and Perlman

(1977), -rr(xiLc) -<c -rr(yiLc)

is equivalent to that there exi. t finite

ao(= -rr(xiLc)),al,···,ap(= -rr(yiLc))

such that

ai-l -<cd; a

ⁱ ^{for some}^di^E

Ac,

ⁱ⁼^1,^·^·

·,

^p.

For each

ai.,

^let^bibe a point belonging to

(Lc

⁺^£

( _g )

₊

_x)

_n

_H�

_n

Sllxll

(23)

and satisfying -rr(bil Lc) ⁼ai. (bi is unique.

)

Then, obviously bi-1 -<ccd,.g) b2. Th<' invPr e is obvious from trasi ti,·i ty of -<c G,g) ^sinceXi-1 -<cedi ,g) x; ^impliesxi -1 -<c r;,g) Xi. ^I

From the next lemma, we know that the ordering -<cc,g) is a special case of the cone ordering S:. K introduced in Section

1.3.

Lemma 1.10 For x, ^y^E^Sr,x -<cc,g) ^yis equivalent to x S:.K(G,g) ^y, ^where

K(G, g)= U C(d, -d, ^g).

dEAc

Proof From the definition, x S:.K(G,g) y is equivalent to that there exist finite x^o(= x),x1, ^{· · ·}

,

xp(= y

)

such that

Therefore, from Lemma

1.9,

it is sufficient to prove the case where

G

=

Gd.

Since

J\(Gd,g)

=

C(d, -d, g)

is convex, x S:.K(Gd,g) ^y^isequivalent to that

(1.8)

_llxll₌

IIYII

^{and y-}^x^E

C(d, -d, ^g).

Setting x ⁼

x1d ⁺ ^x:2g

⁺^z^andy=

y1d ^{+ y2g +}

^z^{for some}^z^E

.C(d, g)..l ^(1.8)

is equivalent to that

(1.9) x1 + 2 x ² 2

⁼

^Yl 2

⁺

^Y2 2

^and

x ²

^:S

^Y2.

vVe notice that 1r(x!Lcd) =

x1d

^andx ^E

Ht

(x ^E

Hi)

is equivalent to

x ²

^{2: 0}

^(x2

^{:S 0}^resp.

⁾

^.

Since

x1d

-<cd

Y1d

is equivalent to

l x 1

l ^:S

I Y

1

I

,

(1.9)

is equivalent to that one of the conditions of Definition

1.

7 holds. Therefore, the proof is complete. I

(24)

Chapter ²

Monotonicity properties of

the power functions of LRTs for

cone-restricted hypotheses of normal means

2.1 Introduction

Let

X1, X2,

^{· · ·}

, Xn (n

^�

2)

be a sample from a k-dimensional normal population

JVk(J-L,

^o-2

Ik),

where Ik is the k ^xk identity matrix. For a closed convex cone C and a linear sub pace L _included in

C,

we consider the following three hypotheses:

We assume that

. C

and

L

are proper subsets of Rkand

C

respectively. The hypothesis H1 is a generalization of order restricted hypotheses. The study of likelihood ratio tests

(

LRTs

)

for

plays a central role in theory of order restricted statistical inference

(

cf. Robertson, \Vright and Dykstra

(1988)).

Putting

1 n

Y

^=-

:Lxi

and

n

^i=l

test statistics of the LRTs are given as follows:

Ho

vs.

H1-Ho l ^x61

⁼

jj1r(YjC)- 1r(YjL)jj2 E2 01-

^_

jj1r(YjC)- 1r(YjL)jj2

^S⁺

JJY- 1r(YjC)jj2

(

known

o-2)

(

unknown ^o-2

)

(25)

where

l!xll2

⁼

x'x

and

7r(xiA.)

is the projection of

x

onto a closed convex. et A ,,·it h

respect

to

II · II·

See Raubcrtas, �ordheim and Lee

( 1 986)

for the derivation of the e test ^�tatist irs. From

') -2 2 -2

now on, we call the four LRTs, in turn, x01-test, £01-test, x12-test and E 1rtest respectively.

In this chapter, vve discuss the behavior of the power functions of the LRTs "·ith respect

to ^J.L. In this introduction, for simplicity of the argument, we suppo�e that L ₌

{ 0}

^and^C

has no linear subspace except

{0}

unless specified otherwise.

(

Note that the assumptions are not essential in the study of the power functions of the LRTs as seen in Section

2.-L)

\Ve nov,·

consider a decmnposi tion of the parameter ^{J.L E}R k into

and 8 ₌

n;rr·

^J.L

.6. represents the distance from the origin and 8 represents the direction from the origin. The behavior of the power function when � or 8 is fixed has been studied by many authors and their results indicate that the power functions of the LRTs have monotonicity properties.

The monotonicity with respect to .6. when 8 is fixed is discussed in connection with un

biasedness of the tests. Concerning the power functions of the LRTs, a unified argument is presented in Section 2.6 of Robertson et al.

(1988).

Recently Hu and vYright

(199-1)

improved the argument in Robertson et al.

(1988)

and established a further monotonicity

property

of the power function of th�� £01-test.

-2

In the restricted testing problem, to understand the behavior with respect to 8 "·hen 6 is fixed is one of 1lte important subjects. The first result on the monotonicity with respect to 8 i ^· found in Barthvlomew

(1961).

He studied the LRT for homogeneity of the components of the mean ^J.Lagainst order restricted alternatives. Restating his result in the case ^\Vhere L =

{ 0},

the power function increases as 8 approaches from each of the edges to the center when k ₌2. It is noteworthy that this monotonicity property does not depend on .6. :::: 0. In addition, he gave some conjectures for higher dimensional cases. See also Section 2.5 of Robertson et al.

(1988)

for these conjectures.

After Bartholomew's study, many authors discussed the monotonicity with respect to 8.

However, most of these studies are numerical. In higher dimensional cases, we ha,·p obtained no

explicit expressions of the power functions which are useful for analytical studies

(

^cf. Singh and Schell

(1992),

Singh, Schell and Wright

(1993)

and Singh and vVright

(1989)).

0.Iathematical results for higbcr di1nensional cases are given only for some special cones.

Oosterhoff (1969)

treated the cast wh1· ·e C is the positive orthant and Pincus

(1975)

treated th case

where

^Cis

(26)

a circular cone.

Section 2.2 gives fundamental monotonicity properties. The monotonicity

results

are deri\'C.··d

m a unified manner by using probability inequalities obtained in Chapter

1.

These results indicate that similarities of the behaviors of the power functions between the

I61-test

and the E

�

1-test and between the x

i

2-test and the E

i

2-test. Sections 2.3 and 2.4 give precise studies of the monotonicity with respect to 6. In Section 2.3, we consider the case where the cone C is symmetric. The monotonicity with respect to 6 proved by Bartholomew

(1961)

implies that the power function is unimodal with respect to 8. \Vhen k ⁼2,

C

is symmetric with respect to a reflection. \Vc extend this unimodality result for higher dimensional cases. In Section 2.-±, we treat the case where

C

is an order restricted cone. Conjectures given by Bartholomew

(1961)

and by Robertson et. al.

(1988),

which specify configurations of the alternatives yielding the maximal power or the minimal power, are considered. We succeed in proving that some of the conjectures are correct.

2.2 The fundamental monotonicity results

In this and the next sections, we suppose that L

= { 0}.

When L

= {0},

noting that

IIYII2 = 117r(YIC)II2

+

IIY-7r(YIC)II2,

the rejection regions of the LRTs with a critical value

c

2 0 are represented as follows:

tbe x

�

1-test:

the E

�

1-test:

{(Y, S) I IIYII2 -117r(YIC)II2

>

cS}

Note that

Y

_and

S

are independently distributed as N

(

JL,

(52 Ik)

and (52

X�(n-l)

respectively.

n n

For q E R, we d<'fine

T(x; q, C)= llxll2- qllx-7r(xiC)II2·

We begin with inveEt igations of the properties of the function

T(x; q, C).

Lemma 2.1

T (x;

rz.

C) is a convex fuction of x if q :S 1.

Lemma 2.2 Suppose q > 0

and

K

is a closed cone. For a closed conve.x cone

C, the

followin.a

are equivalent.

(27)

(1) T(x;

q,

C)

is increasing in �K.

(2) C

is increasing in �K.

Proof

FordE Rk, let

[(1)

^=>

(2)]

\Ve assume that Cis not increasing in �K, which implies that for some dE

J(,

there exists

x

^E

Cd

such that

rd(x) �C.

Then, since

llxll2

=

llrd(x)ll2

^and

we have for

q

> 0

T(x; q, C) llxll2- qllx- 7r(xiC)II2

>

llrd(x)ll2- qllrd(x)- 7r(rd(x)IC)II2

T(rd(x);

q,

C).

Hence

T(x; q, C)

is not increasing in �K·

[(2)

^=>

(1)]

From Lemma

1.6,

it is enough to show that for any fixed d E

J(

and

x

E

C d , T(x; q,

C

)

�

T(rd(x); q, C),

equivalently

(2.1)

Since

llx- Yll2

⁼

llrd(x)- rd(Y)II2

^and

ct

^u

cd

is symmetric with respect to

rd,

we have

(2.2)

Moreover, because X E

Hd

^and

c�

^c

Ht'

(2.3)

for ally E

C�.

Since

c

⁼

(CJ

^u

C�i)

^u

c�,

noting that

llx- 7r(xiC)II2

^=min

llx- Yll2,

we obtain

(2.1)

^from

yEC

(2.2)

^and

(2.3) .

^I

For

C,

^define

J((C)

=

{

dE Rk

I rd(x)

E

C

for all

x

E

C

ⁿ

Hd}.

[-

t

o

,

to]}=

Convcd(x). Since

7r(xtiC) +(to - t)d

�

C,

we have

Since llxtll2

�

llxll2

f

_or

t

E

[-to, to],

_{for any}q 2:: 0, we obtain from

(2.-i)

This implies that -T(x; q,

C)

is light-tailed for

C. (2)

follows from Lernma

2.1

and Corollary

2.1.

I

Characterization of J(

(C)

vVe shall give son1e results on characterization of

K(C).

vVe first notice that

K(C)

is a closed cone.

Lemma 2.4 For a closed convex cone

C, K(C) �C.

Proof Since Ic(x), the indicator function of

C,

is increasing for

C:

the result follows from Lemma

1.8.

I

Lemma 2.5 Fur a closed convex cone

C, K(C) = _K(C*).

Proof Suppose that

d

E

f{( C).

Similarly to the proof of Lemma

2.2,

for

d E

R k, vve consider the decomposition

Cci

⁼

C n Hci,Ct =

{rd

(

^x

)

I X E

Cd'}

^and

c� =

{x E

c

I X�

ct UCd'}·

Fix any x E

C*�1-.

Nf'te that x1y 2:: 0 for ally E

C.

^{Since x}

= xo- tod

and rd

(

^x

) =

xo

+ tod

for some x

o

E

.C(d)

l. ar1d

to

2:: 0, we have

I I

dl

I

dl

I

( )

X Y = x0y

- to

_y

�

x0y +

to

Y

=

Y r d X

(29)

for any y E

H"t.

Since

C�

^C

H"t,

we have (2.5)

lVIoreover, by the symmetry of

c:

^u

Cct

with respect to rd , we have (2.6)

From (2.5) and (2.6). infy'rd(x) 2:

0,

which implies rd(x) E

C*.

Therefore dE

A-(C*),

that is

yEC

K(C)

^c

K(C*) . I<(C)

:J

I<(C*)

is obvious from

(C) =C.

I

From Lemmas 2. 4: and 2.5, we have

K(C)

^:J

C

ⁿ

C*.

Lemma 2.6

If C is a circular cone, i.e. for some

dE

51 and 0 �

t:

� 1

C =

{x

^{E Rk}

I

d'x

2

cjjxjj}

then K(C) ⁼ Ht.

Proof Fix any x ^E

H"t

andy E

H;;

ⁿC. We suppose that jjxl/

= 1

without loss of generality.

Then, since rx(y)

=

y-2(x'y)x, we have

d'rx(Y) ⁼d'y-2(x'y)(d'x).

Since d'x 2::

0

and x'y

� 0,

from the above equation

d'rx(Y) 2: d'y 2: t:jjyjj

=

t:jjrx(y)jj , which implies rx(Y) E

C.

^I

The monotonicity results From now on, we donote by

the power functions of the :X61, E

�

1, :Xi2 and E

i

2-tests respectively.

(30)

Theorem 2.1

As

a

function of J-L, {1) -f3o1(J-L) is light-tailed for I\( C), {2) -{3[11 (J-L, o-2) is liyht-tailed for C,

{3) -f312(J-L) and -f3i2(J-L, o-2) are light-tailed for -K(C).

Proof (1)

f3o1 (J-L)

is equal to

where A=

{x

^E

Rk I T(x;

1,

C)� c}.

From Lemma 2.3 (1), A is light-tailed for

I\( C).

Hence the assertion follows from Theorem 1.2.

(2)

{3[11 (J-L, o-2)

is equal to the expectation of

0"2

1

-

Pr

{

^Y^E^B(S)

^I

^Y^rv

Nk(J-L,-)}

n

where B(s) ⁼

{x

^E

Rk I T(x;

1 ⁺

c, C)� c

s

}

for s 2: 0, with respect to S. From Lemma 2.3 (2), B

(

s) is light-tailed for

C.

Since the distribution of S is not depend on

J-L,

we have the assertion from Theorem 1.2.

(3) Noting that

117r(xiC)jj2

⁼

l!x- 1r(xj-C*)ll2

(

cf. Stoer and \Nitzgall (1970)), it is proved by similar arguments to (1) and (2) that

-f312(J-L)

and

-f3i2(J-L,

^a-7·)are light-tailed for

-K(C*).

Therefore, the result follows from Lemma 2.5. ^I

Now we shall consider the monotonicity with respect to 6. =

IIJ-Lll

C

whenever

C

^C

H!.

Another mono tonicity property obtained from Theorem 2.1 will be discussed at the end of Section 2.4.

Theorem 2.2

As

a

function of

^J-L,

{i) f3o1 (J-L) and 1J[J1 (J-L, o-2) are increasing in

�K(C)

and

{ii) f312(J-L) and !3i2(�-t,o-2) are decreasing in

�K(C)·

(31)

Proof

The proof is similar to the proof of Theorem 2.1 by replacing Lemma 2.3 and Theorem 1.2 by Corollary 2.1 and Theorem 1.1 respectively. vVe note that

^t

hr increasing property in

'S.K(-C*)

is equivalent to the decreasing property in

'S.K(C)· ^I

Remark (1) The monotonicity properties of Theorem 2.2 are included in those of TlH�orem 2.1 except in the case of {3[11 (J.L, 0"2).

(2) It is easily seen that the power functions of the E � ctest and the E� 2-test depend on

^1-L

and

0"2 only through ()

⁼

J.L/

^O".

Therefore, we can replace

^1-L

with () in the above two theorems.

In the next two s<:>ctions, we mainly discuss the monotonicity with respect to

8

of

/301

(J.L) and

!3cn (J.L, 0"2). These results are derived from Theorem 2.2.

Remark Concerning the power functions of the tests for H1 against H2- H1, it may be reason

'S.K(C)

on �1. On the other hand, on �2, the ordering

<c

defined by

X <c y {:::=::> y

-

^X^E

C

is useful. It i!3 known that the power functions are decreasing for C (cf. Theorem 2.6.2 of Robertson et al. (19�8 )) ^.

2.3 An implication of the symmetry of the cone

vVhen

k =

2, the monotonicity with respect to

8

mentioned in Section 2.1 implies that the

power function is unimodal with respect to

8.

We note that C is symmetric with respect to a reflection when

k =

2. In this section, we develop a general argument regarding the relation between unimodality vvith respect to

8

and symmetry of C in higher dimensional cases.

vVhen

k =

2, C

⁼ ^C

(a1, a2), where a1, a2

^E

51 are linearly independent, is symmetric with respect to

ra1-a2

不等式制約を持つ正規分布の平均の統計的推測に関 する研究