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Tests for bivariate mean residual lifeKanwar Sen a & Madhu Bala Jain aa Department of Statistics , University of Delhi , Delhi, 110007Published online: 27 Jun 2007.

To cite this article: Kanwar Sen & Madhu Bala Jain (1991) Tests for bivariate mean residual life,Communications in Statistics - Theory and Methods, 20:8, 2549-2558, DOI: 10.1080/03610929108830649

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COMMUN. S T A T I S T . - T H E O R Y M E T H . , 2 0 ( 8 ) , 2 5 4 9 - 2 5 5 8 ( 1 9 9 1 )

TESTS FOR BIVARIATE MEAN RESIDUAL LIFE

Kanwar Sen and Madhu Bala Jain

Department of Statistics University of Delhi

Delhi-110007

Key Vords and Phrases : Survival function; bivariate exponentialjty; BDPfRL distributions; BNBUE; consistency; asymptotic normality; U-statistics.

ABSTRACT

In this paper we have developed tests for bivariate

exponentiality against the 'bivariate decreasing mean

residual life (BDMRL)' and 'bivariate new better than

used in expectation (BNBUE)' classes of non-exponential

probability distributions. We have also obtained a large-

sample approximation to make the test readily applicable.

1. I NTRODUCT I ON

Let X and Y denote the survival times of two devices

having a joint distribution function F ( x , y ) with

F(o,o) = 1, where F ( x , y ) is the joint. survival function

o f these two devices and is given by

F(x,Y) = P [ X > x , Y > y ] . I n re1 iabil ity and 1 ife-testing

analysis, it is of interest to test H o : F is exponantial

against the various hypotheses describing positive aging.

The we1 1 known classes of 1 ife distributions based on

positive aging are : increasing failure rate ( I F R ) ,

Copyright C3 199 1 by Marcel Dekker, Inc.

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25 50 SEN A N D JAIN

increasing failure rate average ( IFRA), new better than

used (NBU), new better than used in expectation (NBUE),

decreasing mean residual life (DMRL) and harmonic new

better than used in expectation (HNBUE). For def in - - A a t r u details of these classes see for example, Ho!

and Proschan (1984) and Rolski (1975).

Multivariate versions of IFR, IFRA, NBU, NBUE

and HNBUE have been developed by Esary and Ma

(l974), (1979), Buchnan and Singpurwal la (1977 1 ,

tions 3 n A a . . - , , L . = L

DMRL

shal l

B 1 ock

and Savits (1981) and Basu, Ebrahimi and Klefsjo (1983).

Some tests have also been proposed for testing :

H, : F is bivariate exponential against

HA : F is bivariate NElU(ENE1J~-I and ENEV-11

by Basu and Ebrahimi (1984) and against

HB : F is bivariate increasing failure rate gverage

by Basu and Habibul lah (1987) and against

HC : F is bivariate harmonic new better than used in

expectation (BHNBUE)

by Kanwar Sen and Jain (1990) and against

H D : F is bivariate new better than vsed (FEIEU)

by Kanwar Sen and Jain !?991).

To the best of our knowledge the l iterature does not

contain any development of tests far bivariate decreasing

mean residual life and bivariate new better than used in

expectation a1 ternatives. The purpose of the present

investigation is to develop such tests. Here we consider

two aspects.

! i ) In section 2 we derive a test statistic fur testing

bivar iate exponential ity versus decreasing mean

residual l ife alternatives.

(ii) In section 3 we derive a test of bivariate

exponentiality versus bivariate new better than

used in expectation alternatives.

The asymptotic behaviour and the consistency of

the test statistics are discussed in section 4. In

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TESTS FOR BIVARIATE MEAN RESIDUAL LIFE 2551

section 5 we illustrate the techniques of section 2 and

3 with an example.

2. A TEST FOR B IVARIATE DECREASING HEAN RESIDUAL LIFE ALTERNAT!VES

The bivariate mean residual life function of (x,y)

(See Buchnan and Singpurwalla (1977)) is defined as

In this section we consider the prublem of testing

where ?(x,y) is the sl~rvival function for the

bivariate exponential distribution introduced by Marshal 1

and Olkin (1967).

versus

HI : Bp(x,y) 2 B p 0 ( x , y ) for a1 I x and y uith strict

inequality for at least one x and y on the basis of a

random sample ( X i , YI! . [ X 2 , Y2', . . . iK,, Y,: of size n from the

distribution F.

Consider

- - c ~ ~ ( x + t , y + t ) d t Po ( x , Y)

We introduce the following measure from the null

hypothesis H I , for iife distributi~n F b a l o n p i n < f a H I :

A (PI = [ 7 7 ' P ( x + C, y+ C) d~dxdy 0 0 0

where Z = min(x,y) 1 2 . 4 )

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2552 SEN AND JAIN

U n d e r H,,, A ( F ) = 0 a n d ! u n d e r t ( l , A (F) > 0.

Now by t h e c l a s s i c a l n o n p a r a r n e t r i c a p p r o a c h o f

r e p l a c i n g E ( X Y ) , E t X ) , E ( Y ) , E(z 'Y I . E l 1 2 ~ ) , E ! % . i l ) s o d

i -1 T h e n D, i s a n e s t i m a t e n f A (F). H u n c e t h s p r o p v , z e '

t e s t f o r t e s t i n g H o ? ? . ? i n s t H I i s :

R e j e c t H o i n f a v n ~ u r of H i i i t ~ , d , 11: a c i i t i i - 1

v a l u e su t h a t t h e t e s t i s o f size r .

3. A TEST FOR BIVARIATE NEU BETTER THAN USED IN EXPECTATION ALTERNATIVES

Tk:e f n ! ! c u i n g d e f i n i t i o n o f h i v a r i a t e new h e t t e r

t h a n u s e d i n e x p e c t a t i o n a p p e a r s i n E l ~ c h n a n a n d

S i n g p u r w a l l a ( 1 9 7 7

D e f i n i t i o n 3. 1 : F i s a i d t s b e E?!ifl~'E i f

0 r e - l x , y ! 5 e F ! C , s i

i . e . i f f n r a l l t 2 C) t h e b i v a r i a t e m e a n r e s i d l ~ a i l i f e a t

t i m e t i s n o t g r e a t e r t h a n t h e m e a n r e s i d u a l 1 i f e a t t i m e

11. T h e b o u n d a r y m e m b e r s nt ENEIJE c l a s s o b t a i n e d w h e n

t h e r e i s i n e q u a l i t y i n ( 3 . 1 ) a r e t h e b i v a r i a t e

e x p ~ ~ n e n t i a l d i s t r i b t ~ t i i i n s (EVE) n f M a r s h a l l a n d 0 l i : i r - i

i 1967 I .

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TESTS FOR BIVARIATE MEAN RESIDUAL L I F E 2553

We consider the problem of testing H o a= given in

(2.2) versus

distribution F. Let

Now viewing y (F) as a measure of deviation f rorn Ho,

the classical nonparametric approach of replacing

- ~ ( x , y ) , E i X ! , E ( Y ) and E ( 7 ) by P n ( x r y ) ,

1 n -

and Tz ~i respectively, where F , ( x , Y ) = No and n N ( x , y ) is the number of ( X i , Y i l ' = such that X i ) x , Y i i

y and Z i = minix 1 1 ., Y . ; . suggests rejecting H o in favour of

r 1 i f a > b tta,bl = <

l o otherwise It is more convenient to reject H o against H 2 for

large values of the asymptotically equivalent statistic

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2554 SEN AND J A I N

4. PROPERTIES OF D, A N D T,

In this section we study the asymptotic hehaviour of

the test statistics D n 3n.d T,. Since b n t h sre ! I -

statistics, we have following theorems

;fi ( D ~ - A [Fl ! Theorem 4.1 : T h e f r i 1 -.:.ing distr i b : > t inn of

a is normal with mean zero and variance unity, where

and

where

and 1 T,j = -jjz ( X ~ + Y J

T = ( T , , T,, . . . , T 6 ) , 8 = (0 , . 8 , . . - . .8,) = E ( T ) and

Corollary 4.1 : !!rider the n:iI i h y p ~ t i a ~ i s I;', is

asymptntical ly normal with mean A(F) . The variance of

f i ( ~ , ) is a function of X I , A, and ALa.

Theorem 4.2 : The 1 in1 t i n g distr i b ~ u t i n n of f i ( T n - y ( F ) )

is normal with mean zero and variance

where

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TESTS FOR R l V A R I A T E MEAN RESIDUAL L I F E

and = cov(U) = ( o i l ) , i = 1,2,3,4 j = 1,2,3,4

- . Corollary 4.2 : l in r ' . . i i t h e . t , i n I r -

asymptoticaiiy normal

f iTn is a f unct inn of

Since the exprrs5 i

under H , 3 r 2 :'STY C U Z ~ !

with mean y ( F ) T h e variance of

I1. I, and A,,.

nn for t h e v a r lances of f iLln andfiTa

lc.31 ei and 5 l c r p i . t h 3173 Tn ?re

functions of W s e a t i s t i c s . w e use iacLknif ing to estimate

the variances of f i D n and fiT, are

f i T n are asymptotica1iy standard nnrmai and hence Q($~T,) I

D, 3nd 7 , a r u !cansistent.

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2556 SEN AND JAIN

TABLE I lntravascular albumin pool before and after prednisolone

Intravascular Albumin Pool ( g )

Patient Before. X i After. Y i Z

i 7 4 . 4 8 3 . 8 7 4 . 4

5. AN EXAMPLE

Cain, Mayer, and Jones (1970) have studied albumin

and f ihr inogen met-abol ism by using the carbcnate-pl asma

proteins before and after a 1 3 - d a s course of

prednisolone. The eight subjects were patient, : , i t _h

hepatoceiluiar disease as established by needie biopsy.

Part of stl~dy related ta changes in the intravascuiar

albumin pool. Tabie I is basej u n a subset of the Cain-

Maycr-Jones data.

Using the data in Tat!e I . we have computed D,, 2nd

T". T b g ras~-!! t % 3 1 s <i : . en hel.2~:

Since Z 1 s n i i Z 2 3 r s s t s n d i d i?;;..;mal ;q? ;s je . : t fi, , x f

E ' JE in favuur of buth H i 3 p d Yy

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TESTS FOR BIVARIATE MEAN RESIDUAL L I F E

BIBLIOGRAPHY

Basu, A.P., Ebrahimi, N. and Klefsjo, B . (1983). Mu1 tivariate HNBUE distribution. Scand. J. Statist. 10, 19-25.

Basu, A.P. and Ebrahimi, N. (1984). Testing whether survival function is bivariate new Setter than used. Commun. Statist. (Theory and Methods l3( l5), 1839-1849.

Basu, A.P. and Habibul lah, M. (1987). A test for bivariate exponentiality against BIFRA allternative. Calcutta Statistical Association Bulletin. 36, 79-84.

Block, H.W. and Savits, T.H. (1981). Multivariate classes in reliability theory. Mathematics of Operations Research. 6, 453-461.

Buchnan, W.B. and Singpurwalla, N.D. ti977). Some stochastic characterization of multivariatesurvival. Theory and Applications of Re1 iabil ity, Vol. 1, 329-348, Ed. C.P. Tsokos and I.N. Shimi, Academic Pbess.

Cain, G.D., Mayer, G. and Jones, E. A. (:l970). Augmentation of albumin but not fibrinogen synthesis by corticusteroids in patients with hepatocel lular disease. J. C l in. Invest. 49, 2198-2204 ( 1970).

Esary, J . D . and Marshal I , A. W. : ( 1974). Mu1 tivariate distributions with exponential minimums. Ann. Stat. 2, 84- 98.

Esary, J . D. and Marshal I, A. W. ( 1979). Mu1 tivariate distributions with increasing hazard rate average. Ann. Prob. 7, 359-370.

Hol lander, M. and Proschan. F. (1984). Nonparametric Concepts and Methods in Re1 iabil ity. Handbook of Statistics, Vol. 4, Nonparametric Methods, Ed. P.'R. Krishnaih and P . K . Sen, 6 13-655.

Kanwar Sen and Jain, M.B. (1990). test for bivariate exponential i ty against BHNBUE al ternative. Commn. Statist. (Theory and Methods) A19(5), 1827-1835.

Kanwar Sen and Jain, M.B. (1991 ) . A new test for bivariate distributions : exponential vs new-better-than-used alternative. To appear in Coomn Statist. (Theory and Methods).

Marshall, A.W. and Olkin, I. (P967). A multivariate exponential distribution. J. Arne,r. Stat. Assoc. 62, 30- 44.

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2558 SEN AND J A I N

Rolski, T. ( 1 9 7 5 ) . Mean residual life. Bulletin u f the International Statistical Institute. 46. 3 6 6 - 2 7 1 2 .

Sen, P.K. 9 Some invariance principles relating to jackknifing and their role in sequential analysis. Ann. Stat. 5, 316-329.

Received December 1990; Rev ised A p r i l 1991.

Recommended Anonymously.

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