COMPARING THE VARIABILITY OF RANDOM-VARIABLES AND POINT-PROCESSES

In this paper we compare the variance of functions of random variables and functionals of point processes. Specifically we give sufficient c onditions on two random variables X and Y under which the variances va r f(X) and var f(Y) of the function f of these random variables can be compared. For example we show that if X is smaller than Yin the shift ed-up mean residual life and in the usual stochastic order, then var f (X) less than or equal to var f(Y) for all increasing convex functions f, whenever these variances are well defined. In the context of point processes we compare the variances var phi(M) and var phi(N) of the f unctional phi of two point processes M = {M(t), t greater than or equa l to 0} and N = {N(t), t greater than or equal to 0}. We provide suffi cient conditions under which these variances can be compared. Specific ally we consider comparisons between (i) two renewal processes and bet ween (ii) two (homogeneous or nonhomogeneous) Poisson processes, For e xample we show that for any nonhomogeneous Poisson process N with a ra te function bounded from above by A and from below by mu, and two homo geneous Poisson processes L and M with rates A and mu, respectively, v ar phi(L) less than or equal to var phi(N) less than or equal to var p hi(M) for any functional phi that is increasing directionally convex i n the event times, whenever these variances are well defined. This, fo r example, implies that if T-n is the nth event time of N, then n/lamb da(2) less than or equal to var(T-n) less than or equal to n/mu(2). So me applications of these results are given.