The rank of the current lifetime

Statistics & Probability Letters 20 (1994) 269-271

ELSEVIER

The rank of the current lifetime Rudolf Department

Griibel

of Mathematics and Computer Science, University qf Paderborn, 33095 Paderborn. Germany Received September 1993

Abstract Consider the standard model of renewal theory, where components with independent and identically distributed lifetimes are installed successively. We obtain the asymptotic distribution of the relative rank of the lifetime of the component in use at time t as t + cc. The result exhibits another aspect of the classical inspection paradox.

Keywords; Lifetime; Rank; Limit distribution

We consider

the standard

renewal

up: x1, x2, . . . are independent variables (the lifetimes), all with tion

theoretic

set-

positive random distribution func-

F, nENo:

N,=max i

exist i=l

I

is the number of renewals up to and including time t and L, = XN, + 1 is the total lifetime of the component (historically, a lightbulb) in use at time t. The rank of the current lifetime is then given by R, = #{I

< i < N,: Xi > L,},

R,/N, is its relative rank. We assume that the mean ,U = EXi = jxF(dx) of the lifetimes is finite. Feller (1971) is the standard reference for renewal theory and its applications. In this note we obtain the asymptotic distribution of the relative rank of the current lifetime. Let us briefly recall the easier case where F is continuous and where the Xi’s are compared in a more 0167-7152/94/$7.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0167-7152(93)E0183-T

direct manner: the rank of X, + 1 within the values X 1, ... > X, is uniformly distributed on (0,1, . , n}, as is immediate from a permutation argument. Dividing this rank by y1 and letting IZ-+ co we obtain a non-trivial limit distribution, the uniform distribution on the unit interval. In the renewal theoretic case, however, XN, + 1 and X1, . . . , XN, are no longer independent; also, it is well known that the distribution of L, will in general be different from the original lifetime distribution. The question of interest to us is: how do dependence and lengthbiasing affect the distribution of the relative rank of the current lifetime?

Theorem. The ratio R,/N, converges in distribution as t -+ CC; the distribution function given by

G(x) =

s

1 P

[F-1(1

P(dy) > -x),00)

G of the limit is

270

R. G$~bel I Statistics & Probabjlity Letters 20 /t994/ 269-271

where F-‘(x) = inf{y: F(y) 3 x}. Also, G(x) 3 xfir allx,O
1.

Proof. Let fin(x) = n-l Cy=, lte,xr(Xi) be the empirical distribution function associated with x1, . . . , X,. The following simple observation is the key to the prooE R,/N, =

1 - &&).

< x) =

”

P

G(x) 3 3

s

lim P(L, < x) = P(L,

> 1 - F(F-‘(1 3 x. (21

< x)

1 - F(L,) + F(L,) - &<(L,).

Let E, := SUP,~~I F,,(x) - F(x)]; E, converges to 0 almost surely as n -+ co by the ~livenko-CantelIi theorem. As N, f co almost surely with t + co we also have Ed,-+ 0 almost surely; to obtain the limit distribution for R,/N, it is therefore enough to consider 1 - F(L,). Using (2) and

we

*

F(x) & y

get

P(l - F(L,) d x) = P(F(L,) 3 1 - x) = P(L, 2

+ P(L, as desired.

-x)-)

c1

< x),

for all x E 1w,not just the continuity points of the limiting distribution function. This holds in the lattice case and in the non-lattice case. The usual distinction between the two cases is not necessary here: if the distribution of Xi is concentrated on multiples of some h > 0 then R,/N, remains constant on intervals [kh, (k + l)h), k E mi,. From (1) we have

x >, F-‘(y)

- x) - )

F(dy) s [O,F-‘(lpx))

yF(dy).

*+cO

R, - = N,

1 - F(F-‘(I

G(x) 3 1 -

10,x1

lim P(Lt d x) = P(L,

WY) --x),m)

SimiIarly, for F- ‘( 1 - x) > y,

Indeed, we even have

t+m

1

J[F-‘cI

L x.

(1)

It is a classical result from renewal theory that L, converges in distribution to L, where P(L,

To obtain G(x) 3 x we consider two cases: if x is such that F- “( 1 - x) 3 p then y/p > 1 for all y with F-‘(1 - x) < y, hence

F-‘(1 - x))

2 F-‘(1 - x)) = G(x)

If F is continuous then we can rewrite the distribution function of the limit as G(x)

=

s

1 CL

1

F-‘(y)dy;

l-s

this function is closely related to the Lorenz curve associated with F (see Goldie (1977) and C&go et al. (1986)). Also, the behaviour of R,/t is immediate from the theorem and the fact that N,/t converges to l/p almost surely. The last part of the theorem shows that there is a decrease in distribution as we pass from the direct to the renewal case, an observation that can be interpreted as a feature of the inspection paradox. It is also obvious from the proof that the effect of dependence vanishes as t tends to infinity and that length-biasing takes over: the same limit would arise if we compared Xi, . . . , X, to L with X ii . . . , X,, L independent. In general weak convergence does not imply convergence of moments, but here it does as 0 d R, d N,. In particular, for continuous F, lim Es = 1 - xF(x)F(dx) L--‘cc N, s which is less than or equal to 4 by the above order considerations. Example. If the Xi’s are exponential with mean p (which means that we have a Poisson process

R. Griibel 1 Statistics

& Probability

with constant intensity) then some straightforward calculations show that the limit distribution has a density g given by g(x) = - logx, 0 < x < 1; the associated expectation is $. These do not depend on p as is also obvious from a resealing argument. Note that the moment of order - 1 of the limit distribution is infinite. Finally we mention that the above method seems to be applicable in the finite-mean case only. If 1 - F(x) is regularly varying at + co with index - a, 0 < a < 1, then R,itself converges in distribution. This has recently been shown by Scheffer (1993) using point process methods.

Acknowledgement I would like to thank Charles Goldie for stimulating comments on an earlier version of this note.

Letters 20 (1994) 269-271

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References Csiirg& M., S. Csiirgs and L. HorvBth (1986), An Asymptotic Theoryfor Empirical Reliability and Concentration Processes, Lecture Notes in Statistics No. 33 (Springer, Berlin). Feller, W. (1971), An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 2nd ed.). Goldie, C.M. (1977), Convergence theorems for empirical Lorenz curves and their inverses, Adu. in Appl. Probab. 9, 765-791. Scheffer, C.L. (1993), The rank of the present excursion, Stoch. Process. Appl., to appear.