On multivariate mean remaining life functions

JOURNAL

OF MULTIVARIAfE

ANALYSIS

On Multivariate

25, 1-9 (1988)

Mean

Remaining

Life Functions

BARRY C. ARNOLD Department

of Statistics,

University

of California,

Riverside,

California

92502

AND

HASSANZAHEDI* Department

of Mathematics, University of California, Santa Barbara, California 93106 Communicated

by P. R. Krishnaiah

The concept of the mean remaining life function is extended to the multivariate case. Some general characterization properties of multivariate mean remaning life functions (mmrl) are studied, it is shown that they determine the joint pdf uniquely. Inter alia a basic convergence result is proved for a sequence of mmrl functions. The multivariate loss of memory property of a pdf is characterized in terms of the mmrl function. The relationship between the mmrl function and the hazard-gradient is discussed. i(-l 1988 Academic Press. Inc.

1. INTRODUCTION

In recent years considerable attention has been paid to the problem of characterizing the probability distribution function (pdf) F of a random variable (T.v.) X based on conditional expectation in general, and in particular on its mean remaining life (mrl) function. See, for example, Shanbhag [26], Hamdan [16], Kotlarski [18], Funk [lo, 111, Swartz [27], Dallas [9], Laurent [20], Arnold [l], Gupta [15], Nagaraja [24], Kotz and Shanbhag [19], and Bhattacharjee [S]. It is natural to ask if there is an analog in the multivariate case. As far as we know there have been two instances where multivariate extensions of the univariate mrl Received August 1982; revised June 29, 1984. AMS 1980 subject classifications: Primary G2H05; Secondary 62ElO. Key words and phrases: Conditional expectation, multivariate mean remaining life function, multivariate distributions, stability theorem, loss of memory property, multivariate exponential distributions. l Present address: Department of Statistics, Florida International Park, Miami, Florida 33199.

University, University

1 0047-259X/88 $3.00 Copyright 0 1988 by Academic Pms, Inc. All rights of reproduction in any form reserved.

2

ARNOLD AND ZAHEDI

function have been implicitly Buchanan and Singpurwalla

defined. The most general form proposed by in [S 3 is

g(x)= som P(X> x + t) dt/P(X> x),

x > 0;

(1)

(their others are special cases of (I )>. Ghosh and Ebrahimi [ 141 also made use of g(x) as defined in (1). Although g(x) seems a reasonable, direct extension of the univariate mrl function y*(x), y*(x) = E(X-

x 1x> x) = 1” P(X> t) dt/P(x> .x

x),

P(X>x)>O;

nevertheless, it does not have the most essential property of the univariate mrl function. That is, it does not determine the corresponding pdf uniquely. In the present note we supply a definition of a mean remaining life function that does characterize the corresponding pdf uniquely.

2. A MULTIVARIATE

MEAN

REMAINING

LIFE FUNCTION

Let F be the joint pdf of a random vector X on RP (p 2 1). The extended real valued vector b = (b,, bZ, .... bp) is called the supremum of the support (SOS) of Fif bj = infix: F,(x)

where F,, is the ith univariate marginal an empty set is defined to be co.

= I),

distribution

(2)

of F, and inlimum

of

2.1. DEFINITION. Let X = (X,, X2, .... ,I’,) be a random-vector on Rp with joint pdf F. Let b= (b,, b,, .... bp) denote the SOS of F, Xi+ = max(O, Xi) and suppose

E(X,+)< Define a vector-valued

00

for

i = 1, 2, .... p.

(3)

Bore1 measurable function y(t) on RP by

y(t)= (r,(t),yz(t),... Y,(t))= ax - t I x 2 t),

(4)

for all t E RP, t < b such that P(X b t) > 0. The vector-valued function y(t) will be called the multivariate mean remaining life function (mmrl). On the set {t: P(X 2 t) = 0}, the function arbitrary fashion.

y(t) can be defined in an

MULTIVARIATE

MEAN

REMAINING

LIFE

3

FUNCTIONS

The following notational convention will be used repeatedly. For any scaler w and any vector t E RP-’ the symbol (wcij, t) will denote the vector we li+l, *..Y t). (t I, ..-Y ti--l, Suppose x has SOS b and mmrl y(x). Let Ai denote the set t )) exists and equals 01, where t E RP -I. Then for {Z: lim.ytz(sw YAx(i,7 i = 1, 2, .... p, h-= ’

inf{Z: ZEAi} i 02

if if

Ai#@ Ai=@.

(5)

The proof parallels that provided by Kotz and Shanbhag [19] in the univariate case. A convenient representation for the mmrl is provided by, for i = 1, 2, .... p, 1 Xat]

yi(t)=E[Xi-ti

=g-J

Ftx+

(i)’

t,

dx,

(6)

l,

where t
The proof parallels that given by Kotz and Shanbhag in the univariate case (note that we condition on X > t rather than Xi > ti). Equation (6) shows that yi(t) is the ratio of two left continuous functions and hence itself is left continuous. Clearly t is a point of continuity of F if and only if it is a point of continuity of yi (i = 1, 2, .... p). It is evident from (6) that knowledge of the mmrl of X is sufficient to determine the mmrl of any subcollection of the x;s (by letting superfluous t;s converge to - co). Conversely we can express F as a function of y as follows. For any s
..-Y

fi-

13

siv

Si+

1 y ...T

Sp)

yi(t*,...,fi~I,ti,Si+l,...,Sp)exp

(8)

and F(t) = lim s---m

Yitfl,

fi

.*9

ti-19

si3

si+lv

..*,

sp)

..7

ri-l,

ri,

si+l,

-a*?

Sp)

ev

i=l

Yitr

1,

du ( t13 ...? ti- 1, 4 si+ 17...y Sp)

where -m=(-co,...,

-CO)‘.

(9)

4

ARNOLD

AND

ZAHEDI

Note that (8) is a consequence of the fact that for any i, -

du s 51Ydt I,...? fi--l~UISi+It-~~~Sp) 18

= log

[

Ydt

17 ...Y ti>

Yitt

1)

. ..)

sj+

tj-

I? ...? sp)

1)

Sit

o..y

Sp)

Qtl,

..-7 ti,

F(t,

3 ..*)

tip

si+ 1,

1, . ..p Sp) Siy

3e.y

Sp)

1 ’

i = 1, 2, ...) p. 2.2. Remarks. 1. From (5), (6), (8), and (9) it follows that knowing the mmrl function y(t) for all t (-toe < t < b) determines the corresponding pdf F uniquely, and, vice versa. This is consistent with the univariate case. If in addition we assume that P has a continuous gradient vF= (aiyax,, .... a@?x,), then it is not difficult to check that Eq. (8) can be written as:

where

t Y;(u)+1 I( s

’

Y 1(u)

Y:(u)+1 . ..’

.*

y,(u)

)

denotes a line integral, and is independent of the path (piecewise smooth), and Y;(U) zdef (a&) Y,(U). 2. X has independent marginals if and only if yitt)

=

Y,Y,(ti)

Vi, Vt < b.

A stability result, analogous to the uninvariate

case is given by 2.3.

Let {F” : n = 1,2, . .. } be a sequence of pdf s on RP, with a 2.3. THEOREM. corresponding sequence of mmrl functions (y”: n = 1, 2, ...}. respectively. Let F be a pdf on RP with corresponding mmrl functions y and supremum of support b. Suppose that these two conditions hold: (i) (ii)

lim sup”+ m jllX,, , T dF,(x) -+ 0 as T-+ co (tightness condition) Fq(x)O,

where g,‘s are bore1 measurable functions, integrable with respect to Lebesgue measure on R’; Fq(x) = 1 - Fg(x-), where Fq denotes the pdf qf the ith

MULTIVARIATE MEAN REMAINING LIFE FUNCTIONS

5

univariate marginal of F,,, i = 1, 2, .... p. Let C be the set of continuity points x < b of F (and consequently of y). Then Y”(X) --, Y(X)

as

n-+oO,

VxEC

(i.e., y;(x) + yj(x), i = 1, 2, .... p) if and only if

I;,(x) --*F(x)

as

n-too,

VxEC,

i.e., F, +w F.

The proof is a straightforward generalization of the univariate proof. It is not difficult to construct examples to show that the convergence theorem (2.3) need not be true if either condition (i) or (ii) is relaxed. For instance, consider the two examples of Kotz and Shanbhag [19]. Note. An alternative candidate mmrl function is y*(x)= E(x - t ) X > t). If F is continuous then y E y*. In general y(t 1+ , .... t,+ ) = r*tt I, .... tp) so that y* determines the corresponding pdf uniquely as does y.

Almost

all our results for y remain valid for y* if we replace F(x) by

P(X > x) whenever necessary.

3. RELATIONSHIP BETWEEN THE MMRL

FUNCTION AND THE

HAZARD-GRADIENT

In the context of reliability theory, in analyzing multivariate survival data from an absolutely continuous pdf F on P”+ , the hazard-gradient plays a major role. 3.1. DEFINITION. Let X = (Xi, . ... X,) be a random-vector the joint survival function S,

on RP, with

S(x 1>.“, XJ = P(X, > x,, ...) x, > x,), where x = (xi, .... xP) is in RP, . Let R(x) = -log S(x) for x in the set (x: S(x) > O}. The function R(x) is called the hazardfunction (analogous to the univariate case). If R has a gradient r =VR, then t is called the multivariate hazard rate or vector multivariate hazard rate or hazard-gradient. More explicitly, if for x E {x: S(x) > 0}, i = 1, 2, .... p, ri(x)=~,R(x), I

Then r(x) = (rl(x),

.... rP(x)) is the hazard-gradient.

The hazard-gradient

6

ARNOLDANDZAHEDI

has been discussed by Block [6,7], Johnson and Kotz [ 171, Marshall [21,22], and Barlow and Proschan [3,4]. If F is a pdf on RP, with SOS b and such that VF exists. Then the corresponding mmrl y and hazard-gradient r are related by r,(t) = (WQA y(t) + 1 I

rdt)

’

i=l,2

,..., p,

(11)

t
This follows, using the continuity and differentiability of F, from differentiating the expression for y,(t) with respect to ti and noting that r,(t)

=$

I

i= 13 .... P.

~tt)/~tt),

Expression ( 11) and our earlier results can be used to rederive Marshall’s [21] assertion that the hazard gradient uniquely determines the pdf.

4. A CHARACTERIZATION OF THE MULTIVARIATE Loss OF MEMORY PROPERTY

We begin this section with the following definition: 4.1. DEFINITION. A pdf F on RP is said to have multivariate memory property (MLMP) if P(X, > XI + A, ...) X,>x,+A

loss of

X, > xp) = P(X, > A, .... X, > A),

1 xl>xl,...,

or equivalently, F(xl + A +, .. .. x,+A+)=~(x,

+ ,..., x,+)&A+

,..., A+).

for all x,, .... xp and A. When p = 1 (p = 2) we use LMP (BLMP) MLMP.

(12)

instead of

If F has the MLMP it is not difficult to verify that F(O+, . ... 0+ ) = 1 and that F is continuous in each variable. One can thus rewrite (12) as qx,

+ A, x2 + A, .... xp + A) = F(x,,

x2, ... x,)F(A,

vx,, x2, ...xp.

A very simple characterization

of the MLMP

A, ... A) ALO,

is available in terms of the

MULTIVARIATE

mmrl function. That is, for F continuous only if Yi(t+A

‘ItERP,

7

MEAN REMAINING LIFE FUNCTIONS

on RP,, F has the MLMP

if and

‘l)=Yi(t),

and

VA>0

for

i=l,2

,..., p.

(13)

The proof of this assertion follows from (8) and (10) (with s=O) using the definition of MLMP. Remarks. pdf F. If X follows that sequently X*

(i) X = (Xi, .... X,) be a random vector on RP, with joint (or, equivalently, F) has the MLMP, then immediately it X* = min i S iSp Xi also has the LMP property, and conis exponentially distributed.

(ii) Let F be the pdf of a random vector X on RP, with the MLMP, and corresponding mmrl function y = (yi, y2, .... y,). Obviously y,(t . 1) = ~~(0.1) = Ci, a positive constant, Vt 2 0, where Cj= E(X,), i= 1, 2, .... p. Nevertheless the condition ri(t. 1) = Ci, Vt >,O and for i= 1, ,..,p neither implies nor is implied from the condition that X* = mini S- i Z - p Xi is exponentially distributed. For example, consider F(.wz, =+exp(--x,-x,)+$exp-max(x,,x,)x,,x,>O. (XI.XZl and

5. AN EXAMPLE

Consider the sdf F, given by I, + A2- Ai)-‘(A, e1,

x2) =

exp[ -(A, + A2 - A;) xi - (A, + 1;) XJ

+ (2, - W w( -Jx,)),

Xl

5

x2,

Xl

L x2,

(~,+I,-I;)-‘{I,exp[-(IZ,+I;)x,-(3,,+I,-I;)x,] + (A-Wexp(-k)j,

(14) where the parameters A,, I,, I,, &, A; are all positive, i = 1, + A, + A,, x1, x2 > 0. This distribution, which generalizes both the bivariate exponential distribution (BVE) of Marshall and Olkin [23] and the BVE of Freund [12] was introduced by Proschan and Sullo [25] as an example

8

ARNOLD

AND

ZAHEDI

for some sampling and inference results. The above survival function can be found in Friday [13], where the distribution is referred to as the PSE (Proschan-Sullo extension) distribution. The first component of the mmrl function y corresponding to PSE distribution is given by

Yl(t) =

4 bwCW2 - 6 )I - 1) + W2 - Mh - b) +e:Cn,+e,(~,-n;)(n,+n;)lre,(n,+nl,)l-’ t, G12, 8,{1,expC8,(t,-t,)l+1,-I;} ’ J-,(1,+ W evC&(h - b)l + (4 -M& + 4 + A-’ 7 t, 2 f2, & expCW, - f2)1+ 4 -4 (15)

where 8, = A, + A2 - ,I; and t12= 2, + 1, - 2;. The expression for y2(t) is obtained by interchanging the roles of the subscripts 1 and 2 in (15). From these expressions it is evident that the PSE has the BLMP. The marginal mean remaining life functions yx,(t) = yI(t, 0) and yx,(t) = ~~(0, t) are clearly not constant so that the distribution given by (14) does not have exponential marginals. For further examples see Arnold and Zahedi [2]. ACKNOWLEDGMENT

The authors thank the referee for helpful suggestions.

REFERENCES

[l] [Z] [3] [4] [S] [6] [7] [8]

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MULTIVARIATE

[9]

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[16] [ 171 [ 183

[19] [20] [21] [22] C23] [24] [25]

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9

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