滋賀大学学術情報リポジトリ

起

Tests f

o

r

Exponentiality

under Random Censorship Model

Y

o

s

h

i

k

i

K

umazawa

S

h

i

g

a

U

n

i

v

e

r

s

i

t

y

与11\~

1

<

%

Tests f

o

r

Exponentiality

under Random Censorship Model

Y

o

s

h

i

k

i

Kumazawa

S

h

i

g

a

U

n

i

v

e

r

s

i

t

y

Abstract

The aim of this thesis is to present classes of test statistics for testing ex -ponentiality against some alternatives with an aging property under a random censorship model. The exponential distribution is widely used in the fields of reliability

，

survival analysis and life testing because of its simple nature. The feature of the exponential distribution can be described as constant failure rate function or constant mean residuallife function. This property presents a good description of the life length of a unit which dose not age with time. But there are some situations that the occurrences of the initial failures and the wearout failures cause the changes of the failure rate function and the mean residuallife function. Such aging properties give rise to the correspondings for the life distri -bution. By using the concepts of six nonparametric models for life distributions with the aging property

，

we consider six testing problems under the random censorship model.

Censored data arise naturally in many fields. The underlying test may be a destructive one so that units on test can not be re-used or

，

because of time and or cost constraint

，

we can not afford to wait indefinitely for all the units to fail. And as in a clinical trial

，

patients may enter the study at different times and leave

，

or die from a cause different from the one under investigation. Depending on the nature of the under1ying tests

，

some types of censored data may be found

，

and we deal with the censored data observed under the random censorship model.

In Chapter 2 we give these notions on life distribu tions and types of censor -ing and we review in Chapter 3 some basic results from the theory of counting processes

，

martingale limit theorem and von Mises statistical functionals. Based on these mathematical foundations

，

we give an asymptotic theory of all proposed

1.1. ABSTRACT

statistics presented in Chapters 4-6 with unified approach.

Each of the sections of these chapters discusses different testing problems under the random censorship model and proposes some classes of test statis -tics based on the Kaplan-Meier estimator. The asymptotic distribution of all proposals is derived under the null hypothesis and五xeddistributions. And a consistent estimator of the asymptotic variance of each statistic under the null hypothesis is construded from the theory of counting processes. The compari-son of the tests on the basis of the Pitman asymptotic efficacy is also given for some alternatives under the proportional censoring model and we recommend one test from this result for each testing problem.

Contents

1

Introdnction

2

Lue Distribntions and Types of Censoring

2.1 Life Distributions ... 5 2.2 Types of Censoring ... 8

3

Connting Processes and yon Mises Fnnctionals

1

0

3.1 Counting Processes ... 10 3.2 Von Mises Fundionals ... 14

4

Tests for IFR姐 dIFRA

1

6

4.1 The IFR Alternative ... 16 4.2 The IFRA Alternative ... 23

5

Tests for NBU and NBUE

3

4

5.1 The NBU Alternative ... 34 5.2 The NBUE Alternative ... 39

6

Tests for DMRL and HNBUE

5

0

6.1 The DMRL Alternative

.

.

.

.

.

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50 6.2 The HNBUE Alternative ... 57 Acbowledgments References

6

7

6

8

Chapter

1

Introduction

The exponential distribution is as widely used in reliability and life testing as the normal distribution is in other areas of statistics. One of the reasons is that the mathematics associated with the exponential distribution is relatively simple. In this respect the "lack-of-memory" property that the remaining life of a used unit whose life time is represented by exponential variable is independent of its initial age plays a central role. This property can be expressed by a simple functional equation of the life distribution function (df) and presents a good description of the life length of a unit which does not age with time. But there are some situations that the occurrences of the initial failures and the wearout failures cause the changes of the failure rate function and the mean residual function

，

and such aging properties give rise to correspondings for the lifedf.

In Section 2.1 of Chapter 2 we define a variety of life distributions ac -cording to aging properties represented in nonparametric forms. These are: the

mcreαsing failure rate(IFR)

，

theincreasi吋 failurerateα附 、age(IFRA)

，

the

neωbetter than used(NBU)

，

thenew better thαn used in ezpedation(NBUE)

，

thedec何 αSt句 meαnresidual life(DMRL) and theharmonic nω better than used in ezpectation(HNBUE) classes. Each of the notions of aging has a simple statistical interpretation and has a dual property by reversing the inequality or the direction of monotonicity. These are named DFR

，

DFRA

，

NWU

，

NWUE

，

IMRL and HNWUE

，

respectively. The IFR distribution has a increasing failure rate function and the IFRA distribution has a increasing failure rate average. The NBU property states tha.t the conditiona.l surviva.l proba.bility ofa.used but

2 CHAPTER 1.INTRODUCTION

unfailed unit at any age is less than or equal to the corresponding probab出ty

of a new unit and the IFRA class is contained in the NBU class. The NBUE property is a weaker version than the NBU property

，

and says that the expected life length of a new unit is greater than or equal to the expected remaining life of a used but unfailed unit at any age. The DMRL distribution has a decreasing mean residuallife function and the corresponding class contains the IFRA class

，

but is contained in the NBUE class. The notion of the HNBUE property may be interpreted出 statingthat the integral harmonic mean value of the mean

residual life function at any age is less than or equal to the integral harmonic mean value of a new unit. The class of the HNBUE life distributions contains the above :five classes and may be considered as a more natural class of the life d

f

'

s.In this thesis we consider these life distribution classes as the alternatives for testing exponentia五ty.

Section 2.2 of Chapter 2 de:fines three types of censoring:Type 1

，

Type II and random censoring. Censoring is often occurred in survival analysis and life testing

，

and the censored sample contain only partial information about a population of interest. In Chapters 4・6the problem of testing exponentiality is

considered under the random censorship model.

In Section 3.1 of Chapter 3 we present the theory of counting process and martingale limit theorem in connection with the treatment of the censored data observed under the random censorship model.In the censored case we make sta -tistical iI由rencesabout the population distribution

F

(

t

)

or about some function -als of

F

(

t

)

by the use of the Kaplan-Meier estimator

F

.

.

π，

(

t

)

.

This Kaplan-Meier

estimator may be considered as a generalization of the usual empirical df in the uncensored case. This fact together with the theory of counting processes and martingale limit theorem helps us to discuss the behavior of statistics based on the Kaplan-Meier estimator in the uncensored case

，

as wel1as in the censored case

，

with a uni:fied approach. The normalized process of this estimator is known to be expressed as the stochastic integral with respect to the martingale gener -ated from the censored data and this fact plays an important role in deriving the asymptotics of al1the test statistics proposed in this thesis.

INTRODUCTION 3 of the empirical df

，

von Mises (1947) proposed a technique based on a form of Taylor expansion involving the derivatives of the functionals. In Section 3.2 of Chapter 3 we review the approach presented in Fernholz (1983) based on the Hadamard differentiability of the functionals. Kumazawa (1984

，

1986b

，

1986e

，

1987a) used her method to derive the asymptotic be}即 iorof the statistics rep -resented as a functional of the Kaplan-Meier estimator

F

n

(

t

)

.

In Section 4.1 of Chapter 4 the problem of testing exponentiality against the IFR alternatives is considered. The test statistic is consbucted by using the prope均 ofthe scaled total time on test (TTT-) transforms 再 l

(

t

)and was

discussed in Kumazawa (1988b). The concept of the scaled TTT-transforms introduced by Barlow and Campo (1975) has proven to be very useful in the statistical analysis of reliability and life testing. The asymptotic distributions of the suitably normalized version of the statistic under the null hypothesis and 五xedalternatives are given. The efficacy consideration of the test for some IFR alternatives under the proportional censoring model is also presented and it is shown that the efficacy decreases with the value of the expected proportion of observing the censored data.

We present in Section 4.2 of Chapter 4 two classes of test statistics for test -ing against the IFRA alternatives. The first class was proposed by K umazawa (1984) and includes the class of the statistics given by Deshpande (1983) in the uncensored case. The second one is considered as a generalization of the statis -tic introduced in Kumazawa (1988b) that utilized the property of the TTT・ transforms. The asymptotic distribution of the proposed statistics is derived and the comparison of the tests is made on the basis of the Pitman asymptotic efficacy.

Section 5.1 of Chapter 5 deals with the testing problem against the NBU alten凶 ives. In Kumazawa (1987a) the Kaplan-Meier estimator was used to

generalize the class of the statistics proposed in Koul (1978a). The asymptotic distribution of the statistic with a weight function is shown and the efficacies of the statistics for some alternatives are computed. From the numerical evaluation of these efficacies one test is recommended.

4 CHAPTER 1. INTRODUCTION

For testing against the NBUE alternatives we present in Section 5.2 of Chapter 5 three statistics N，t N₂ and N_{3 •} The statistics N₁ and N₂ are con

-structed by the same method as given in the previous and were proposed by Kumazawa (1986a

，

1986e). In De Souza Borges

，

Proschan and Rodrigues (1984) the N₁-statistic with constant weight function was considered in the uncensored case. And theN2・・statisticwith constant weight function was considered by Hol

-lander and Proscl

(1980) i血nthe censored case with a modi五edKaplan-Meier estimator. The N

3-statistic is represented as a Kolmogorov-Smirnov type and was introduced in Kumazawa (1989b). For the五rsttwo statisticsN1 and N2 the asymptotic dis

-tribution is found to be normal

，

but the asymptotic distribution of the third one is shown to be not normal.So the efficacy comparison of the tests is made between theN1-and N2・tests

，

and we recommend the use of the test based on

the one member of the class of the _N2-statistics.

In Section 6.1 of Chapter 6 two tests for exponentiality against the DMRL alternatives are given. The proposed test statistics P₁ and P₂ are constructed from the two measures of exponentiality towards DMRL-ness. The

.

n

rst measure is based on the property of the scaled TTT-transforms and the second one uses the notion of the de

.

n

nition. The P1-statistic is a generalized version of the

one introduced by Kumazawa (1988b) and the class ofthe P:γstatistics contains the one proposed in Bergman and Klefsjる(1989)

，

in which a modi五e dKaplan-Meier estimator was used and the proof given in there seems complicated. The asymptotic distributions of the statistics are given and the efficacies of the tests against some alternatives are presented to select the optimal test for the testing problem.

Finally

，

we consider in Section 6.2 of Chapter 6 the problem of testing exponentiality against the HNBUE alternatives. Bergman and Kle色jる(1985) introduced the test statisticsQl and Q2 based on the modi五edKaplan-Meier estimator and Kumazawa (1989a) proposed the Qs-statistics for this testing problem. We present proo色 onthe asymptotic distributions of these statistics

by the theory of counting processes and martingale limit theorem. The efficacy consideration.shows the use of the one member of the Q2-statistics.

Chapter 2

L

i

f

e

Distributions

and Types of Censoring

2.1. Life Distributions We formulate a variety oflife distributions based on notions of aging

，

which afford nonparametric statisticians an opportunity to consider inferences accord -ing to their probabilistic and geometrical properties. DEFINITION 2.1.1. A五島 distributionF(t) isa probability distribution satis -fying F(t) 0 for t

<

O. The corresponding survival function is given by S(t):= F(t):= 1 -F(t). The function

r

t dF(s) A(t):=

_ん

I

l-F(s-) 、 、 . ， f 噌 a A 噌 E 4 9 u ， ， •• _{‘ 、} is called the hazard function associated with F( t).

Note that when F(t) has a densityf(t) and S(t)

>

0

，

dA(t) f(t)

一 一 -λ

(

t

)

dt 1 -F(t-)

is referred to as the failure rate function. Here we may interpretλ(t)dt as the probab出tythat a unit alive at timet will fail in [t

，

t

+

dt)

，

where dt is small.

6 CHAPTER !.LIFE DISTRIBUTIONS AND TYPES OF CENSORING For a discussion oflife distribution classes

，

we need the following notations: a v b

s

/ / し 山 口 汀 噌 E A a v b < F ト 勺 ハ り 一 J E e ， ， I

、

。

F

り

到

し

し

+

∞

f

o

o

M

m

i J S ﹃ 1 一一一一一一 F

一

F λ り μ 7 r h v 九 DEFINITION 2.1.2. (α)F( t) is increa6i吋 failurerate (IFR) if Ft(s)おdecreasingin t E [0

，

TF)ゐr each5

>

O. (b) F(t) is inCrea6ing

i

f

のlurerate averαge (IFRA) if A(t)jt isincreasingin t

ε[

O

，

TF).

(c)F(t) is new better thαn u6ed (NBU) if Ft(s)

三

F(s)ゐrall t and s

ε

[O

，

TF ). (d) F(t) is new better than u6ed in ezpectatio

川

NBUE)if F(t)has a finite

mean μF and μF

三

eF(t)forall t

ε[0

，

TF)

，

where

(

J

ア

S( s ) ds / S (t ) if S ( t)

>

0

，

eF(t):= { ~t

l

0 otherwise

，

1 ._，r 向 λ M 噌 E A 9 u f t

、

and is called themean residua1五feat aget.

(e)F(t) is dec陀 α6mgmeαn re6iduallife (DMRL) ifF(t) has a五nitemean μF

and eF(t) isdecreasingin all t

ε[

0

，

TF).

(

f

)

F(t) is harmonic nω better than u6ed in ezpectation (HNBUE) ifF(t) has

a finitemean μF and

1 = S(s)ds

三 川

By reversing the inequalities and the directions of rnonotonicity we get the six classes DFR

，

DFRA

，

NWU

，

NWUE

，

IMRL and HNWUE

，

respectively. Here

D=decrea6ing

，

I=increa6ingand W=wor6e.

Different properties ofthe five classes IFR

，

IFRA

，

NBU

，

DMRL and NBUE and their duals were considered by authors such as Marshall and Proschan (1972)

，

Barlow and Proschan (1975)

，

Langberg

，

Leon and Proschan (1980) and Hollander and Proschan (1984). The classes of the HNBUE and HNWUE li長

~.1. LIFE DISTRIBUTIONS 7 distributions were五時tintroduced by Rolski (1975) and investigated by Klefsjる (1981

，

1982b). Here the chain ofimplications holds among these life distribution classes: IFR

ι

DMRL 今 IFRA

=

=

=

>

ニニキ〉

NBU

ι

NBUE

=

=

=

>

HNBUE

In this thesis we consider these life distribution classes as the alternatives for testing exponentiality under the random censorship model defined in the next section.

8 CHAPTER ~. LIFE DISTRIBUTIONS AND TYPES OF CENSORING

2.2. Types of Censoring

Censored data arise naturally in a number of fields

，

particularly in problems of reliability and survival analysis

，

and contain only partial information about the population distribution of interest. We discuss three types of right censoring. To this end

，

letX~

，

X~ ，

.

.

"

X~ be independently

，

泊ide佃ntic叩al均l

with lifedf F(t).

(α)宜ype1 Censori時・ Weassume thatn units are put on test and we terminate

our test at a predetemined time T

，

so that complete information on the 五rstk order statistics

X

，

O.¥

<

X~..\ <・・・ < X

。

(1)一 (2)一 一 (k)

is available. Here the number k is an integer-valued random variable(問)

with

X~L\ _(k)

<

_{一 一}

T

<

X

₍

。

_k+1)・

Each of the remaining unobserved life times is known to be greater than the time T.

(b)宜ypeIICensori時・ Asin Type 1 Censoring

，

πunits are simultaneously

put on test and we terminate our test after a predetermined number (or

企action)offailures are obtained. I

.

n

this case we have complete information on the firstl'observations

X

，

O. ¥

<

X~..\ <・・・ < X

。

(1)ー (2)ー ー か )

and the民mainingobservations are known to be grea the number l'(or1'

/

n

)

is a五xedconstant.

(c) Random Censori時 .LetU1

，

U2γ・1九 beiidwith possibly discontinuous and defective df G( t).Uj is considered as the censori時 timesassociated

with Xt and

G

(

t

)

is referred toasthe censori時 df. Then we can only

observe

(X

j

，

8

d

，

1

三

i

三

n

，

where

Xj := min(

x

t

，

Uj) and

ん

:=1{X:5U‘}

~.~. TYPES OF CENSORING 9

stochastically independent. And in Chapters 4・6we assume that the

popu-lation

d

f

F

(

t

)

is continuous. This random censorsl匂 modelarises in med-ical applications with animal studies or clinical trials. In a clinical trial

，

patients may enter the study at different times: then each is treated with one of several possible therapies. We want to observe their life times

，

but censoring occurs according to loss to follow-up

，

drop-out and termination of the study.

Chapter 3

Counting Processes

and von Mises Functionals

3.1.Counting Processes

Itwas demonstrated by Aalen (1978) how the theory ofmultivariate count -ing processes gives a general framework in which both censored survival data and inhomogenuous Markov processes may be analyzed

，

and how by means of martingale central limit theorem the asymptotic behavior for the one-and the two-sample statistics and generalizations to censored data may be derived. Here we give the results from the theory of multivariate counting processes in connec -tion with the treatment of the random censorship model.

Let (0

，

F

，

P)

，

{九:

t E

[

0，

∞

)

}

be a五xedstochastic basis. A multivari -ate stochastic process

N

(

t

)

=

(N

1

(

t

)

，

N

2

(

t

)

，

・・・

，

N

k

(

t

)

)

de

.

n

ned on the time interval

[

0，∞)

is called a multivariate counting process if each of thek com-ponent processes

N

j

(

t

)

has a sample function which is a right-continuous step function with zero at time zero and with a五nitenumber of jumps

，

each of size +1

，

and if furthermore two different component processes can not jump at the same time. Then by Theorem 1.9 of Meyer (1976)

，

there exist right continuous

，

nondecreasi時， predictable processes

A

j

{

t

)

with zero at time zero such that

are local ma町，工.吋rt

“

tii白n時g伊肝ales

P

戸en凶s叫ato位rof爪

N

i

例

(

t

t

の

)

.

11

3.1. COUNTING PROCESSES

Under the random censorship model described in Section 2.2

，

we can ob -Define stochastic serve the possibly right censored data (Xj

，

ん

)

，

1三t三π. process

N

(

t

)

on

[

0

，∞)

by

N

(

t

)

:

ニ

乞

l

{

x

.

引 at

N

(

t

)

represents the number of the uncensored units observed to failure time t or earlier. By Lemma 2.3 of Gill (1983)

，

、 ‘ E ， J 噌 Ei 唱' A q d ， ， ••

、

M

(

t

)

:=

N

(

t

)

-

l

t

Y

(

s

)

d

(

3

.

1.

2

)

is a square integrable martingale on [0

，

∞

)

，

where

Y

(

t

)

:

=

玄

l{Xぷ}・

Here the process

Y

(

t

)

represents the number ofthe units at risk at time

t

and the function

A

(

t

)

denotes the hazard function associated with the

d

f

F

(

t

)

defined in the equation (2.1.1) of Section 2.1.

Under the random censorship model

，

we make statistical inferences about

d

f

F

(

t

)

or about some functionals of

F

(

t

)

by the use of the Kaplan-Meier estimator

F

n

(

t

)

or the functionals of

F

n

(

t

)

.

The estimator

F

n

(

t

)

was first introduced in Kaplan and Meier (1958) and is defined by the population

(

3

.

1.

3

)

制:=

1 -

S

n

(

t

)

:= 1 -

g

{

1

一割

Note that when we on the basis of the censored data (X

わ

ん

)

，

1

三

i

三

九

・

get a complete sample the Kaplan-Meier estimator

F

n

(

t

)

reduces to the usual empirical

d

f

.

The asymptotic behavior of

F

n

(

t

)

on the whole line is discussed by Gill (1980

，

1983) using the theory of counting processes and martingale central limit theorem.

We present some results necessary to discuss the asymptotic distribution of the test statistics based on the Kaplan-Meier estimator

F

n

(

t

)

in the later

12 CHAPTER 3.COUNTING PROCESSES AND VON MISES FUNCTIONALS chapte弘 Forany processW(t) we de

.

n

ne the stopped process WT(t) by WT(t):= W(T八t) with

T

=

max1<i<n

X

i ・ LEMMA 3.1.1 (Gill (1983)). For allt we have /2Fn

(

t

)

-

F(t) π

(

t

)

:

= η11

(

t

)

=

l

t H

州

(3.1.4)

(

3

.

1.

5

)

where S

一 、

y

J

一

い

↓

一

げ

e e -e A丸 一

S

。 a 唱 A η 一 一 e o n H (3.1.6) and

1

(

8

)

:= l{y(.)>O}・

We denote by D[O

，

t

]

the space of right continuous functions de

.

n

ned on the interval [0

，

t

]

with left limits

，

with the Skorokhod metric topology.

LEMMA 3.1.2 (Gill (1983)).Let h(t) be a nonnegative continuous and nonin

-creasingfunction on the interval [0

，

TH] such that

f

μ

(t)dC(t)<

∞

，

where

r

t _d

_A

₍

₈

₎

C(t):=

1

ん

1-H(8-)' TH := sup{t : H(t)

<

1

}

(

3

.

1.

7

)

(3.1.8) and H(t) := 1 -S(t)G(t). Then the stochastic process白

作 ( 附

f

， 小 問 げ

and

十

n(t)dh

州

3.1. COUNTING PROCESSES 13

convergejointly in D[O

，

T

H

]

weaklyasη→ ∞ to processes

川

)Z(.)

，

か

M

respectively

，

where Z(t) isa Gaussian process withzero mean and covariance function E{Z(s)Z(t)}

=

C(S八t).

14 CHAPTER 3.COUNTING PROCESSES AND VON MISES FUNCTIONALS

3.2. Von Mises Functionals

In order to study the田 ymptoticbehavior of statistics that are functionals of the empirical df

，

von Mises (1947) proposed a technique based on a form of Taylor expansion involving the derivatives of the functionals. The approa s 関entedi担nFer口rnholz(1983) was constructed on the basis of the Hadamard differ -e凶 abilityof the functionals

，

and Kumazawa (1984

，

1986b

，

1986e

，

1987a) used her method to derive the asymptotic behavior of the statistics represented as functionals of the Kaplan-Meier estimator

F

n

(

t

)

de貴I叫 bythe equation (3.1.3). Let T(F) be a functional based on F

ε

1)， a class of df 's. And letV and W be topological vector spaces and

ι

(V

，

W) the set of continuous linear transformations from V to')1¥人 LetS be a class of compact subsets ofV such thatS contains alI singletons

，

and letA be an open set ofV. DEFINITION 3.2.1. A functional T

A

→ W is Hadamard differentiable at F E

A

if there exists T

l

.

-

(

・)ε

L

:

(V

，

W) such that for any KεS Hm T(F

+

tH) -T(F) -T

l

.

-

(tH

2

=

0

t→o t uniformly forH

ε

K. The linear transformation T

l

.

-

(

・)is called the Hadamard derivative ofT(

・

)

at F.

Then the following result folIows from Theorem 4.4.2 of Fernholz (1983).

THEOREM 3.2.2 (Kumazawa (1986b)).Suppose that

F

，

バ

t)is an estimator of population df F(t) such that the stochastic process

{

η

1/2[Fn{F-1(t)} -F{F-1(t)}] : 0

三

t

三

1}converges in D[O

，

l] weakly asn→ ∞ toa continu -ous Gaussianproc田5W(t) with zero mean and continuous covariance白nction

，

where F(t) isa version (possibly stochastic) ofF(t). And suppose that the in

-duced functional'T(g):= T(g 0 F) for9

ε

D[O

，

l] is Hadamard differentiable at

the identity function I(t):= t with derivative'T

H

.

)

and that

3.P-.VON MISES FUNCTIONALS 15

Then we have as π→ ∞

η

1

/

2

{

T

(

F

n

)

-T(F)}

→d

N(O

，

0'

2

)

provided0'2

=

Va

1

'

{

rHW)}

>

O.

Since most statistics such asL-

，

M-and R-statistics can be expressed as Hadamard differentiable iundionals as shown in Fernholz (1983)

，

the above result would help us to derive the asymptotic normality of the statistics. The other forms of statistics based on the Kaplan-Meier estimator were discussed in Kumazawa (1986b).

Chapter

4

Tests f

o

r

IFR and IFRA

4.1. The IFR Alternative

We consider to test the null hypothesis

π

。

:

F

(

t

)

= 1 -exp(

-

t

j

μ) for

t三

o

(μunspecl五ed) against the alternative 冗1 :

F

(

t

)

is IFR

，

but not exponential

，

on the basis of the possibly right censored data (Xi， o，)i 1 ::; i

三

n，de五nedin Sedion 2.2. In analyzing the life distribution classes with the aging properties

，

Barlow and Campo (1975) and Kle色jる(1982a)proved that the different forms of the aging properties can be expressed by their corresponding properties of the scaled total time on test (TTT-) transforms

H

;

l

(

t

)

，

where 制 時 一 J u -包一 S

一

F ハ り 一 μ ' F ︼ 一 F L 2 一 一

再

(4.1.1) ior 0

<

t

<

1.

From the result oi Barlow and Campo (1975) we have that

H

;

l

(

t

)

三 t

for exponential distribution

F

(

t

)

and that

F

(

t

)

is IFR if and only if

H

;

l

(

t

)

4.1. THE IFR ALTERNATIVE 17 considered the measure

1

1

μ

1

₁

γ

1

₁

1 -t ト H

一

→

ソ

γt

_ソ

'

l

_{叫 巧 刑}

₁

_{刊 い (} I白 t包t 8 f~OO

F

(

t

)

S

2

(

t

)

{

1

-2

F

(

t

)

}

d

t

μF

of discrepancy between exponentiality and continuousIFR distribution

，

and proposed the test statistic

K

1 :=

;

1

fA(t)

勾

(

t

)

{

1-2

F

n

(

t

)

}

d

t

μn ( 4.1.2) where

戸

n

.

-

か

(

t

)

( 4.1.3) The statisticK 1 may be considered as a generalization of the test statistic

LLL{k(D

j+ν -

Di)-

ν

(D

川 一

Di)

}

introdl悶 din Kle色jる(1983)under the uncensored model which we have a

com-plete sam ple Y1

，

九

，

・

・

・ ，

Yn from

F

(

t

)

，

where

D fff(jjt)

九

(

t

)

d

t

.

-3

・ 2 :

:

=

1

九州

and く= V L

一

1

一

a

z

一

一 一

九 SinceFn(T)

<

1 almost surely if the largest observationT of theX/s is censored

，

the integral region in defining the statistic K 1 becomes to the貧困te random interval [0

，

T

]

.

Note that we reject the null hypothesis討。 infavor of the alternative冗1for large values of the statistic K 1・

THEOREM 4.1.1 (Kumazawa (1988b)). Suppose

t

b

a

t

t

b

e

d

f

'

s

F

(

t

)

and

G

(

t

)

s

a

t

i

s

f

y

t

b

e

c

o

n

d

i

t

i

o

n

s

1

T H

18 CHAPTER 4.TESTS FOR IFR AND IFRA

and

η1/2 h1(T)

→

o

in probability as π

→∞，

( 4.1.5)

wbere C(t) and TH are defined in tbe equations (3.1.7) and (3.1.8) in Lemma

3.1.2of Section 3.1

，

respectively

，

and

h

川州(

ω

収

の

t

)

イ

∞

S

い

Let tbe function g(t) be ofform rt=1(t一 向 )witbα1 = 0 and any αi

，

2

三

t

三

l.

Tben tbe sequence of tbe rv's

converges in distribution asn

→∞

to a normal rv witb zero mean and variance

、 、 . ， r -a y b -， -， •• ‘ 、

-C

一

JU

一

角 ， a

-、

_BJ 一 、E E F -a v b -I1 ‘ 、 -句 e -L M

一

F

一

μ

一

4F

二

μ

、 、 . E J -a v b -r

・ ・ 、

-A -L H

一

句 a

-μ

一

，

a t -E -f M 叩 一 wbere 内:=

1

00 g{S(t

仰

and ん

(

t

)

:=

1

00 S(

包

)

g

'伊(包)

PROOF: We prove the convergence in distribution of the sequence of the r

、

旬

+

司 E E E E E E B E E -d 引_u

，

a 1 J

'

_μ

、

₁ 1 f J R 山 旬 -t L K

g

、 丸

∞

沢

rl

ん

∞

一

f

J

o

-仰 一 日 げ 制 仙 一 丸

u p

t M 包 ︿ C い / パ ， . 針

A a

r T

f

f

h

u

崎 4 司 4 ， ， r ， ， ， η η a o a ' t ・

=

+

凡

W

to an appropriate normal rv for any real numbers 8 and t according to the

Cramむ-Wold Device. We set

+

包 JU 可 E E E E B B -E E﹂

、

_B E ， 、 、 . . ， ， M 仙 ， ， . . ‘ 、 川 A

s

-6

r d L

、 ‘

J n 3

、

J_， 一

ι

ν

1 J t A 川 町 一 ， ， t_、 、 、 . ， J A

ぬ

い

w

d

A

A

rll-LIt T T r I J o f I

ん

句 必 崎 必 ， ， ， ， ， ， ， ， 唱 A ' A π 明 帥 a e a ι 一 一 -品 V

19

4.1. THE IFR ALTERNATIVE

Then we obtain for some constantM

>

0 by the condition (4.1.5) ぷ

/

2

1

恥 一 九

1

=

内

L

OO g{S(包)}ぬ

+

t

L

oo S(u)d包￨

三

(Mlsl

+

It

l

)

n1/2hdT)

=

op(nO) asn-→cxコ. L J い

H

M

α By applying the formula

H

α

i

-

I

I

b

i

=玄

α

(

1c-b1c)

I

I

to the間九， we have 九

J S J T { F (

包

)-

F

n

(

叫

+t

π

サ

T{F(

包)一えい)

Since the Kaplan-Meier estimator

F

n

(

t

)

is uniformly consistent on the interval

[

0

，

T

]

from the main result of Wang (1987)

，

the Slutsky's Theorem implies that

九 isasymptotically equivalent to 間

1/2Sfv(包)一ね包)}iI1{S(包)一向}E{S(包)一句 }

d

叶

+tn1/2f{F(包)一九州包

=

l

T

ん ( 帆

t

川

where

Z

n

(

u)

=

η1/2{F

耳

(

包

)-

F(

包

)}jS(u)and h.

，

t

{

u):= _sh2(

包

)

+

t_h1(

包

)

.

Hence Lemma 3.1.2 of Section 3.1 together with the condition (4.1.4) yields that

九 →dN

吋

V t M (

包

)

)

asn-→cxコ.

Therefore the desired result follows from Corollary 3.1 of Serfling (1980).

_I

COROLLARY 4.1.2 (Kumazawa (1988b)). Suppose that under the null hypoth-eSls冗。 thecensori昭 dfG(t) satisfies the conditions (4.1.4)and (4.1.5) of

The-20 CHAPTER 4.TESTS FOR IFR AND IFRA

orem 4.1.1. Tnen we nave asn→ ∞

πn

υ

ぺ /1 t μ叫 μF守J wnere σ2 :=

1

00

[

J

L

2

S

(

t

)

-J

L

F

g

{

S

(

t

)

}

]

的 川

This corollary shows that under the null hypothesis冗othe asym ptotic variance of the suitab1y normalized version of the test statistic K 1 de五nedin the equation (4.1.2) is given by

f

仰

)

}

d

C

(

t

)

with

g

(

t

)

:=

t

2

(

1

-t

)

(

2

t

-

1). Since this quantity depends on the unknown parameterμand the censoring

d

f

G

(

t

)

，

we must construct a consistent estimator from the censored observations(Xi

，

ん

)

，

1

三

iざπ. The same method as given in the proof of Lemma 2.4of Kumazawa (1987a) shows that

s

:

イ〆向-州)

is a consistent estimator ofσ2

，

_where

r

t _J(

_包

₎

C

(

t

)

:=π

I

_~rl

_{、 (~r〆、日} dN(包)

and

J

(

包

)

is given in Lemma 3.1.1 of Section 3.1. Hence the asymptotically exact test based on the statisticK1 can be constructed by using this estimatorû~.

Next we compute the efficacies ofthe test statisticK1 against some alterna -tives under the proportional censori_時 mode1where the censoring

d

f

G

(

t

)

is given by

G

(

t

)

₌

Sλ

₍

_t

₎

with censoring parameter入 theva1ue of λhas r民e1at厄ti叩on w叩it山h t

伽hep卯ro凶baめb出耐t句y0山fob刷t句t凶a司叩immgunn 附1

In this situation Corollary 4.1.2 imp1ies μ 1 without 10ss of generality and

o

< λ < 1. And the asymptotic variance of the suitab1y norロmalizedversion of K1 under冗ois given by . ， 4 12 13 6 1 σ

，

:=ーーーー一一一一一一一十一一一一一一一一一一一+一一一一-^ _{7-λ6-λ. 5-λ4-λ} ， 3-λ The following IFR life d

_f

_'

s are considered as the alternative for testing exponen

-4.1. THE IFR ALTERNATIVE 21 tiality. (i) 1 -exp(_te

+

l

)

(Weibull)

，

(u) 1 -exp{ -(t

+

Ot2 _{j2)} (Linear failure rate)}

_，

似)

1

ーは

p[-{t+O(t+e-t

_ー

_リ

(Makeham) and (iv)

I

see-.dsjr(1+0) (Gamma)

，

JO

where t

三

0

，

0 ~ 0 and the null distribution冗owith μ = 1 is obtained when

0=0. Since each of the families{Fe(t)}of the alternatives (i)-(iv) listed in the above satis五esthe conditions (A.l)-(A

.

4

)

given by Kumazawa (1986d)

，

it is seen that the sequence{Fe..(t)}withOn =ω-1/2 and c

>

0 is contiguous to the null dfand that the efficacy of the test statisticK 1 is equal to (dERrK

，

l

l

)2 eff(K

t

}

:=土乙{一訪~I 日~ j(

πV

町 [K

心

where Ee [.]denotes the expedation under thedf Feand Va1'o [.]the variar附

under the null hypothesis冗0・ After some calculations we obtain that for the alternative (i) for (ii) for (iu) and for (iv) ザ

f(K1)=(-;

仙 川 凡

eff(K

t

}

=

(

土

)

2

ju

，

i

24 eff(K

t

}

=

(

土

)

2

j

ペ

60 L _ 3 内 . eff(K1)

=

(

一

一

ln2

+

ー

ln3r~ jσ; 3 2

22 入

。

1 10 1 2 3 4

CHAPTER 4.TESTS FOR IFR AND IFRA

Table 4.1 Efficacies

0

/

the statutic K

1

叫 印 thecen A1ternative (i) (ii) (iii) .72834 .36459 .05833 .66199 .33138 .05302 .42453 .21251 .03400 .30424 .15229 .02437

。

v) .19632 .17843 .11443 .08200 Table 4.1 shows the e慌caciesof the test statisticK 1 for the alternatives

(i)-(iv) and sorne values of the censoring pararneterλThe above reSl山sreveal that the efficacy decreases with the value of λ

4.2. THE IFRA ALTERNATIVE 23

4.2.The IFRA Alternative

For the problem of testing the null hypothesis

冗o:

F

(

t

)

= 1 -exp(

-

t

j

μ) for

t

どo(μunspecl五ed)

versus the alternative

行2 :

F

(

t

)

is IFRA

，

but not exponential

，

we may consider the iollowing two measures of exponentiality against the IFRA d

f

'

s:for nondecreasing function1

T

t

(t)

三

o

and constant

s

> 1

，

.d.1

:

=

わ

dS

仰 印 )

and for nonnegative function 'lt

2

(

t

)

，

ふ・

=

A

∞

'

.

[

1

2

{

S

(

t

)

}

f

;

S

(

包

)

d

u

d

F

(

t

)

一・一一 一 μF where

、

_sJ e e Ju a o 内 z a A u r a e 噌 A r ' ' I 0 9 u e e J u a o 句 z a A V a u 唱_i

r

-- '

o

，

d t a y b 噌 EA 今 a dv 一 一 曹

The first measure .d_.l relies on the fact that

F

(

t

)

is IFRA if and only if for alls>1

s

β

(

t

)三S

(

β

t

)

for any

t三

O. Since the scaled TTT-hansforms

Hi

l

(

t

)

_{defined in the equation (4.1.1) of Sec} -tion 4.1 share the property that

Hi

l

₍

_t

₎

_j

_t

_{is decreasing in}

_t

E [0

，

1] for the IFRA d

f

'

s from Theorem 2.1 ofBarlow and Campo (1975)

，

we may COI凶derthe measure

1

1

1

1

時

;(s)HC(t)

8

t

ψ

2

(

8

)

れ

(

t

)

{

一 一 一 一 一

}

d

t

d

8

J

;

_f

電

2

{

S

(

t

)

}

;

J

S

(

8

)

d

8

d

F

(

t

2

= .d. μF for continuous IFRA d

_f

_'

s with positive weight function

ψ

2

(

t

)

.

24 CHAPTER -，.TESTS FOR IFR AND IFRA

Now from these measures we obtain two classes of test statistics

、 ‘ . ， J ， ， b ， ， •• _{‘ 、} ︿ 凡 JU 可 ︾ ﹄

、..

J a 7 b n μ / t ‘ 、 ︿ 円 、 吋

，

J t 噌 A AV T r I

ん

一 一

、

.

，

r d μ 噌_A ，ψ ， ， ••

、

噌 E A -L (4.2.1 ) and

rW2{

丸

(t)}

J~ 丸(包)dudFn(t)

L2(ψ2):= μ相 ( 4.2.2) by substituting the Kaplan-Meier estimator，F叫(t)for F(t).

Under the uncensored model Deshpande (1983) studied the statistic

Ld

'1/;

1

，

β) withψ

l

(

t

)

t

for the above testing problem. And the statistic

L1(ψ1

，

β) was introduced by Kumazawa (1984) and is a version of the test

statistic proposed by Kumazawa (1983) for testing exponentiality against the NBU alternatives in the uncensored case with βinteg自主 2.We reject the null hypothesisπ

。

infavor of the alternative冗2for small values of L₁ ('1/;1

，

β).

And the statistic L2(れ)is an extended version of the test statistic L2(ψ2)

with ψ2(t)三 constαnt

，

proposed in Kumazawa (1988b)

，

by introducing the

weight function'l/;2(t) in the measure s._2・Inthe uncensored data Klefsjる(1983) investigated the testing problem on the basis of the same property of the scaled TTT-transforms and proposed the test statistic

，

which is seen to be asymptoti -cally equivalent to _L2(

れ)

with'l/;2(t)三 constαntin the uncensored case. Here

we reject冗。 infavor of冗2for large values ofL2(ψ2 ).

For testing against the IFRA alternatives

，

Barlow and Proschan (1969) proposed the test statistics based on the normalized spacings which are general -ized to treat the censored observations under theType II censoring model

，

and proved unbiasedness of the test against the alternatives.

THEOREM 4.2.1 (Kumazawa (1984)). Suppose that the weight function

ゆ

1(t) is continuous and piecewise differentiable with bounded derivatives. And suppose that the df F(t) is absolutely continuous and that the df's F(t) and G(t)satis今 the conditions ∞ < C Ju q u H

f

l

_。

( 4.2.3)

4.2. THE IFRA ALTERNATIVE 25

and

η1/2ψ

{

t

S(T)}S(Tjβ)→

o

inprobab出ty邸 π→∞・ ( 4.2

.

4

)

Then the sequenceof the問 包π1/2{L1(

ψ

1

，

s)-W(F)} convergesindistribution asη→ ∞ toa normal rv B with zeromean and varianceE[B2

，

]

where

W

例

:戸=

1

0 0

わ∞

ψ

仇

川

1

バ

ρ

{

σ

附

例

州

引

S

(

舟附附)リ附} B :=

-

1

0 0 Z(st)S(βt)

ψ

a

柳 川 )

+

1

0 0

仰

)S(tj

州

S(t

附

and Z(t)isthe limiti時 processof the normalized Kaplan-Meier process Zn

(

t

)

given in Lemma 3.1.2of Section3.1.

PROOF: Following Theorem 3.2.2 of Section 3.2

，

we first show that the induced functionalr(g):= W(go F) for9 E D[O

，

1]can be expressed出 acomposition of

Hadamard differentiable transformations. For fixedF( t)

，

ψ1(t)and β

，

we define γ1(g

t

}

(S):= sF-1 0 gt(s)

，

γ2(g1' g2)( s) :=ゆt[1-g10 F{g2(S)}] and γ3(g

t

}

:

=

I

g1(包)d包， JO where g1E D[O

，

1]

，

g2E L1[O

，

1]

，

0

三

8

三

1

，

F-1(s):=inf{t : F(t)

三

s}and gt(s):= inf{t

，

1 : g(t)

三

s}.Then from Propositions6.1.1

，

6.1.2 and 6.1.6 of Fernholz(1983)the above transformationsγ1(

・

)

一

ia(

・

)

are all Hadamard differ -entiable atI(t).Thereforer(g)= i30 γ2{g

，

γ1(g)}is Hadamard differentiable atI(t)by the chain rule of Proposition3.1.2 of Fernholz(1983). N ext we note that η1/21{L

t

{

ψ1

，

s) -W(F)}

一

{W(F;)-W(FT)}I

=

n1/21

ル

t

{

S(向 )}d

仰

)

一

ル

dS(T)}dF(z)I

三

2n1/2

ψ

1{S(T)}S(Tjβ) = op(

旬

。

)

asn→∞・

26 CHAPTER 4.TESTS FOR IFR AND IFRA

Hence Theorem 3.2.2 of Section 3.2 together with some calculations yields the desired result. •

We consider the weight function

仇(包)

= ua _{as a special case.}

COROLLARY 4.2.2 (Kumazawa (1984)). Let

ψd

包)=包

a

，

α

三

1.Suppose that

nnder the nnD hypothesis冗othe censoa時 dfG(t) satis五esthe conditions (4.2.3)

and (4.2

.

4

)

of Theorem 4.2.1. Then が/2{Ll(Ua

，

β)-v-1} converges in distri -bntion as n

→∞

to a normal distribn tion with mean zero and variance

σ2β:= 1 = fa，s{S(t)

μ

C(t)

，

where

fa，β(t) :=

(

α

/

ν

)2

(

s

2t21J - 2βt(β+1)ν/β _ t21J/β]

and v:=α

s+

1.

Because of the dependency of the statistic L

₁

(

U

a

_，

_β)

_，

_{αど 1}

_，

_{β >1}

_，

_{on the}

unknowns μand G(t)

，

we must estimate the asymptotic variance σ2J fEom the observations(Xi

，

8d

，

1

三

i

三

η.To this end

，

we set

え

s:=

1

T

ん

β

山 一 州 )

Then it is seen that

9

3

_，βis a consistent estimator ofσi_，βby the same method as given in Section 4.1.Hence the test rejecting冗_oin favor of冗2for

η1/2{Ll(包a

，

s

)-

v-1}/

九

，

β

<

zηis consistent against all continuousIFRA

alternatives

，

where zηis the η-percentile of a standard normal distribution. Next in order to derive the asymptotic distribution of the statistic L2(

れ

)

，

we discuss the asymptotic distribution of the statistic in the form

f

T

ψ

{Sn

(

t

)

}

:

f

Sn(包)dudFn(t) 九

(ψ):=

.1U T'--'.'-'.JJ U fA'n (4.2.5) where

_ん

isdefined in the equation (4.1.3). Some test statistics proposed in this thesis can be expressed as this form and we apply the following result to investigate their asymptotics.

4.~. THE IFRA ALTERNATIVE 27 continuousand that the df'sF(t)

，

G(t)and the weight functionψ( t) satisfy the conditions

1

00∞

[

ω {

卜

世

伊

仰

刷

例

S

珂

ω

的

仰

υ

(

川

削

州

8_り引

削

_)リ}S_列仰刷

ω

₍

s

_{り 円 )い} π

m

雪

引

{S

珂

(T

η

)

}

→

o

血inp戸roぬba幼b泊出

i

t

けya邸S冗 →

∞

，

1

T H句H

好刷

(

例 附

t

t

り

)

(

4

.

2

.

6

)

(

4

.

2

.

7

)

(

4

.

2

.

8

)

and ね1/2hj(T)→

o

in probabilityasπ→ ∞

(

4

.

2

.

9

)

fori = 1 and 2

，

where

的):=

1

仰

)d叫 ん

(

t

)

イ

∞

仰

包

and ん

(

t

)

:=

1

00

ト

{

S

(

8附 + 曹 伊(

8

恥

Then we have asπ → ∞

が

中

where μψ.-

日

σ

(

8

附

d8

(

4

.

2

.

1

0

)

and σ2 _

.

1

;H{μψh1(t) -μFh2(t)PdC(t) ψ ・一 μF

(

4

.

2

.

1

1

)

PROOF: We first show that the random vector

、 、

B E E F / ' a μ ' 、 ‘ . . ， ， e e ， ， E・ 、 、 ︿ 凡 JU 包 JU 、 ‘ . ， r 包 ， ， . . 、 、 ︿

A

f i l

。

- J

、..

1

・

0 ， ， t -︿ 円 九 r d t A V T r t ' ' ' n u F

μ

a ' b JU 、E_E F a T b ， ， .•

、

︿ 仇 T f 1 1 0 / 1 1 ¥ 角 4 ， ， ， ， 4E ‘ 明 伸

一 一

九 A

28 CHAPTER 4.TESTS FOR IFR AND IFRA

is asymptotically equivalent to

B叫:=

-

(

l

T

h

川

(

t

)

d

M

吋

Th2(

山 川

t

where

M(t)

and

H

n

(

t

)

are defined in the equations(3.1.1)and (3.1.6)in Section 3.1

，

respectively. We set e n w ︿ 凡 JU 包 JU 包 ︿ 円 九 r I

ん

、

_SJ a e ︿ 円 九 g d t ， ψ T f t l o 一 一 時

W

and

W(F)

:=わいい)}

.

1

川

u

d

F

(

8

)

to investigate the asymptotic behavior of the second component of the random vector

A

n

.

For五xed

F

(

t

)

and

ψ

(

t

)

，

we define

γ

1

(

g

)

(

8

)

:

=

F-

1

0

グ

(

8

)

and

γ

官2

メ

(

ω

:戸=

1

1ρ1)

曽

町

υ

(

ト山川

叩

川

1トμ

叫

一

イ

叫

tり朴)(

for

8

ε[0

，

1

]

and

g ε

D[O

，

l

，

]

where

g

*

(

8

)

= inf{t

，

1 :

g

(

t

)三 8

}

.

Since

the transformationsγ1(

_・

₎

and

_i

₂

_(・

₎

are Hadamard differentiable at

I

(

t

)

from Proposition 6.1.1 of Fen恥 lz(1983)

，

the functional

r

(・

)

induced on D[O

，

1] by

r

(

g

)

:=

W(g

0

F)

for

g

E D[O

，

1

]

is Hadamard differentiable at

I

(

t

)

by the chain

rule and the expression that

r

(

g

)

₌

i

2

{

i

1

(

g

)

}

.

Note that the derivative

_τ

f

(

g

)

of

r

(

g

)

at

I

(

t

)

is given by

4..I!.THE IFRA ALTERNATIVE We obtain from the conditions (4.2.7) and (4.2.9)

η

1

/

2

1

{

V

V

叫 - W(P)} -{W(

行)-

W{T(pT)}1 =η1/2

L

O O

判

S

(

s

)

}

'

l

S

(

u

)

d

u

d

叩 )

ニ 川

{

S

(

T

)

}

1

T

S

(

s

)

d

s

+

π1/2

L

O O 曹

{

S

(

s

)}的

)

d

s

三

d

内/片/2

弘い

Mμ

J仰川LFバ珂曹叫

{S

珂貯

S

(

σ

η

例

T

小 )

T

)

=0匂Ip(

い

η0) a邸sη→∞. Therefore the Hadamard differentiability of

r

(

・

)

implies that asη→ ∞ where η

1

/

2

{

V

V

叫 - W(P)} = r

f

(

仇)

+

op(n O )

=ん

(T

間

)

-

l

T h2

(

t

)丸 仰

仇

(

t

)

:=η1/2{F~ 0 P-

1

(

t

)

-p

T ₀

_p

_-

₁

₍

_t

₎

_}

_{for0 ::;}

_t

_:

_:

_;

_1. Hence Remark 2.2 of Gill (1983) yields that ρT η

1

/

2

{

V

V

叫 - W(P)} = -

I

h

2

(

t

)

H

n

(

t)

d

M

(

t

)

+

op(π

。

)

asη→∞・ 29 By applying Lemma 3.1.1 of Section 3.1 to the first component of the random vectorAn' it is seen that

山

{

l

T

え 榊

Therefore the random vectorA耳 isasymptotically equivalent to Bn from Theo-民m 4.4of Billingsley (1968).

Since each component of the random vectorBn is represented as the stochastic integral with respect to the square integrable martingale

M

(

t

)

，

and the functions

h

i

(

t

)

'

s

and the process

H

.

叫

(

t

)

are predictable

，

Theorem 2.1 of

An-dersenet al. (1982) together with the condition (4.2.8) implies thatBn converges in distribution asn→ ∞ to a normal distribution with zero mean vector and

30 CHAPTER -，.TESTS FOR IFR AND IFRA

dispersion matrix {σi，;h9，;9' where

的

j

:

=

f

ん

(t)h;(t)d

Hence we can conclude the proof from Corollary3.3of Ser:si時 (1980)and some

calculations. •

COROLLARY 4.2.4. Suppose that the df 'sF(t)

，

G(t) and the function曹2(t)

satisfy the conditions of Theorem 4.2.3.Then we have asπ→ ∞

九叫ん(れ)-と主主;→

d

N

(

O

，

生

)

，

l μ F J

where 内 : Jand σ~ :J denote the correspondings to those given inthe equations

(4.2.10) and (4.2.11)ofTheorem 4

ユ

3for the function曹2(t)

，

respectively.

COROLLARY 4.2.5.Suppose that under the null hypothesis冗othe censoring df G(t) and the function雪2(t)satisfy the conditions (4.2.6)

イ

4.2.9)of

Theo-rem 4.2.3.Then n1_/2 _L2(

_ψ

₂

₎

_{converges indistribution出 冗 → ∞ toa normal}

distribution with mean zero and variance

σ

ι

わ

2{S

附

2

(

仰)，

where

引 ト

'

l

w2(s)ds

This corollary shows that under the null hypothesis the asymptotic vari -ance σ!:Jof the statisticL₂₍

れ)

defined in the equation(4.2.2)depends on the

unknowns μand G(

t

)

.

Similarly as in the case of the statistic L1 (

ゆ

1

，

s

)

we can

find a consistent estimator

手

:

イ

〆

ω

崎 山 伐

t

)

This helps us to construct the asymptotically exact test based on L2(

ψ

2

)

'

Now we compare the efficacy of the test statistics L1(仇

，

s)

，

αど1

，

s

>

1

，

and L2(

ψ

2

)

for the alternatives (i)-(iv) listed in Section 4.1 under the propor

4.~. THE IFRA ALTERNATIVE 31

Corollaries 4.2.2 and 4.2.5 we may take μ = 1 and 0 <λ< 1.And the asymp-totic variances of the s凶 ablynormalized versions ofthe statistics Ll (-ua

，

β) and L2(ψ2) under 行oare given by

イ:=

s

(

:

r

[

2

a

s

:

1

-

λ+2u-

ふ

+

1)

り 山

and

σ

;

イ

〆

(

t

)

九 ， respectively. Then we have the following efficacies of the statistics Ll

(

u

ぺ

β) against the alter口rna eff{L

t

{

♂，β)} = (αβlnβ

)

2

/

(

v

4

σ

i

)

， for (ii) eff{Ll(

包

a

，

β)}

=

{αβ(β_ 1)}2/(v4σ

i

)

，

for (i

垣

)

α+1 2 α . eff{L

t

{

♂，β)}=

{一一一一一一一一}/イ

v v+1 v +β and for (iv)

(

s

-

l)lnν-αslns eff{L

t

{

包a

，

β)}= 一 2(V -s)2

σ

?

For the test statisticL2(ψ2) we consider the weight function to be of form ψ2(

包)=包

P

，

ρ >-1. Corollary 4.2.5 implies that

σ (

1 B(1-λ

，

2ρ+3l+ ー . 2，ρ ・ー(ρ+2

)

2

¥(ρ

+

1

)

2

(2p + 3

)

2

(1 -λ)I +1ρ(

)

2

+ 4B(l-λ，4p +

7

)

2B(1-，A'p + 2) 一 (2p + 3)2 (p + 1)2(2ρ+ 3)

+

ω

倒

(

υ

1

一

λ

λ

，

勾

い

+

刊

叫

4

的

) 姐倒(

υ

1

ト一

λ

川

3ρ

+

什

叫

5

吟

)

一 . (ωρ+1

り

)(σ2ρ+3

幻

)

2

(ρ+ 1)(2ρ+ 3) }'

32 CHAPTER 4.TESTS FOR IFR AND IFRA Table 4.2 Efficacie$ 01 the IFRA-te$t $tαtutic$ 叫 enthe cen$oring parameterλ =

ふ

αndi A1ternative Statutic (i) (ii) (iii) (i

り

λ= 1/10 L₁₍包1，1.03) 1.2677 0.3168 0.0626 0.4775 L1(包 ¥1.95) 1.2841 0.2985 0.0606 0.4967 L₁₍ u1.5， 3.42) 1.0506 0.1083 0.0329 0.4981 L2(包

0

)

1.1945 0.2404 0.0554 0.4729 L2(包1) 1.0810 0.3594 0.0643 0.5756 入 =3/4 L₁₍包1，1.03) 0.7012 0.1752 0.0346 0.2641 L₁₍包1，1.95) 0.7421 0.1725 0.0348 0.2870 L₁₍包1.5，3.42) 0.8561 0.1600 0.0268 0.4058 L2(包

0

)

0.7545 0.1518 0.0349 0.2987 L2(包1) 0.5184 0.1728 0.0308 0.2760 L2(

旬

。

)

and L2(U1 ).Kumazawa (1988b) discussed the statisticA~ equivalent to L2(uO). Then the asymptotic variances of the s凶 ablynormalized versions of L2(1

川

andL2( u1) under冗。 aregiven by 2 1 2 4 1 1 =一一ー一一一一一一一一一一+一一一一一ー一一一一一+ 2，0 - 9(7ー λ) 3(6-λ) ， 3(5一入) 4-λr 4(3一入) and σ~1=~( _- l+B(1-λ， 52++~B(l 一入， 11)_{一 一一 一} 2，1 -

9

¥

100(1ー λ)1 4 1 1 25 B(1-λ ，3)I 2B(lー λ，6) 2B(1一入，8)¥ 一 一 一 ・ respectively.

4.J!.THE IFRA ALTERNATIVE 33 And the efficacies of

L

₂₍ 11.0) and

L

₂₍ 11.1) are given: for the alternative (i)

eff{L2(J)}=(-;

叫 2

川

/

σ

;

，

。

唱 77_ _ 33 _ _ 2 ‘ -eff{L2(包

l

)

}

=

(一一

_{90 -}l_--n2

+ー

ln3

+ー

ln5)2/σ;，u ， 90 --， 15 for (ii) ef _{f{ L2}( 11.

O

)

}

=

(

土

)

2

/

σ

ふ

24 唱 19

‘-eff{L2(

ピ)}

= (一一 r~

σん

/

1350 for (iu)

e

ザ引州

η

f

削

f

パ

{υωL ‘ 15

‘-ef f{L2(

ピ)}

=

(一一

)2/σ

ん

2520 and for (iv) ' A 角 4 穐 Z H σ ， ， ， ， ， ， ， 句 z a 、 ‘ . . _， ， v h v n ' ・ ・ A h F 1

一

6 J 句 + / / 向 。 由 j n q o z i n d i 一 ほ d ' E -A 4_・ ・ ﹄ -F d ， . q d

+

+

9 B 9 a n n E A E I 唱_i t o h u -h 凸 RU 一 向 ， u 唱_i -a ・_{ . ‘ 向 4 -A H 4

一

一

， ， t 、 ， t t_、 一 一 一 一 -J 1 J 、 •• ，，、 •• ， ， n u 噌 且 制 叫 包 ， ， ・ z_、 ， ， ・ 1 、 今 4 笥 Z M L L r4tr4t t J I J Z J Z J e e respectively. Table 4.2 shows the efficacies of the IFRA-test statistics L1( 11.¥1.03)

，

L1(11.¥1.95)

，

L(包u

，

3

必)

and L2(11.O) for the alternatives listed in Section 4.1

and some values of the censorI時 parameterλ. For the statistic L1 (包a，

s

)

we

choose the values of αand

s

so as to maxImize its efficacy against a particular

alternative. Since 6n :=

1

:

7

=

1

6

d

πis an estimate of P(X1

三

Ud=市

，

we rec・

ommend theL1(11.1

，

1.95)-test statistic for small values of6_n and theL1(11.1

，

1.03)

-test statistic for large values of 6n in the sense of the Pitman asymptotic relative

v o r

e

4 L

P

a

h

c

Tests f

o

r

NBU

and NBUE

5.1.The NBU Alternative

In this section we are interested in testing the null hypothesis

π

。

:

F(t)= 1 -exp(-tjμ) fort ~ 0 (μunspecl五ed) versus the alternative

冗3 : F(t)is NBU

，

but not exponential

，

under the random censorship model.

Koul (1978a) considered the parameter

1

00

1

∞ 附

)}'It

{

S

(

t

)

}

一 州 + り } 川

μ

=

ψ

(

い

仲

8

吋)州一

1

00

1

00 'It

{

S

(

s

+

の

}

d

F

(s

)

d

F

(

t

)

as a measure of the deviation of

F( t

)

Itom exponentiality towards the NBU alternatives and developed the class of the test statistics in the uncensored case. Here the weight function

世

(

・

)

is assumed to be nondecreasing. This parameter with

ψ

(

t

)

=

t

was五rstinvestigated in Hollander and Proschan (1972).

For the testing problem based on the censored observations(Xj

，

8

d

，

1

三

t

三

n

，

Kumazawa (1987a) proposed the class of the statistics

山):=

1

T

1

T

ψ{

_{丸 山 山}

(5.1.1 )

5.1. THE NBU ALTERNATIVE 35

which corresponds to the the Koul's (1978a) NBU statistic in the uncensored case. The statisticM1 (ψ) withψ(t)= t was considered in Chen

，

Hollander and

Langberg (1983a) using a modi五edKaplan-Meier estimator ofF(t).

THEOREM 5.1.1 (Kumazawa (1986c

，

1987a)).Suppose that the weight function ψ(t) is continuous and piecewise di長rentiablewith bounded derivatives.And suppose that the df F(t) is absolutely continuous and that the d

f

'

s F(t) and

G(

t) satisfy the conditions

J

九

2(t)dC(

(

5

.

1.

2

)

and

η1/2ψ{S(T)}→

o

in probability asπ→∞・

(

5

.

1.

3

)

Then the sequence of the問 包πn1/

_刊

2{M₁(

世

利

)

一

W(F

円

)}c∞on附 r堵ge白S 1血ndistribution as η→ ∞ to a normal rv B with zero mean and varianceE[B2

，

]

where

W

例

:戸=

l

co ∞

l

co_∞

ψ

叫

仰

{

σ

似仰

刷

_珂

S

_的

仰

_υ

₍

8+t

り

)

阿

s

り

M

柑刷川

μ

)

川阿附

d

削仰

σF

町

叩

(t B:=

-

l

co

l

co

仰 + 仰

+t

州

8+

什t)}dF

附

0

+21 co ∞ 1 t

ρ

却 -

Z 8)

卯 一 り

dF

山 川

t)}dF(t) and Z(t) is the limiting process of Zn(t) given in Lemma 3.1.2 of Section3.1.

PROOF: To apply Theorem 3.2.2 of Section 3.2

，

we五rstnote that the induced functionalr(g):= W(g 0 F) for 9 E D[O

，

1] can be expressed as a composition

of Hadamard differentiable transformations. For五xedF(t) and ψ(t)

，

we define

and γ1(9

t

}

(

8

)

:= F-1 o9t(8)

，

γ2(92)(8

，

t):= 92(8)

+

92(t)

，

i'a(9a

，

91)(8

，

t):=ゆ

[

1-

910 F{9a(8

，

t)}] 判(93):=

1

1

か い ， 仰

t

，

where 91ε D[O

，

1

，

]

92E L1 _[0

，

₁

_，

_]

_{93ε L}1_[0

，

_{1]x [0}

，

₁

_，

_]

₀

_三

8

，

t

<

1 and