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Comparing the Variability of Random Variables and Point Processes

Published online by Cambridge University Press:  27 July 2009

Mark Brown  and
J. George Shanthikumar
Mark Brown
Affiliation:
Department of Mathematics, The City College, CUNY, New York, New York 10031
J. George Shanthikumar
Affiliation:
Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California, Berkeley, Berkeley, California 94720
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Abstract

In this paper we compare the variance of functions of random variables and functionals of point processes. Specifically we give sufficient conditions on two random variables X and Y under which the variances var 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 Y in the shifted-up mean residual life and in the usual stochastic order, then var f(X) ≤ 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 Φ(M) and var Φ(N) of the functional Φ of two point processes M = {M(t), t ≥ 0} and N = {N(t), t ≥ 0}. We provide sufficient conditions under which these variances can be compared. Specifically we consider comparisons between (i) two renewal processes and between (ii) two (homogeneous or nonhomogeneous) Poisson processes. For example we show that for any nonhomogeneous Poisson process N with a rate function bounded from above by λ and from below by μ and two homogeneous Poisson processes L and M with rates λ and μ, respectively, var Φ (L) ≤ var Φ (N) ≤ var Φ (M) for any functional Φ that is increasing directionally convex in the event times, whenever these variances are well defined. This, for example, implies that if Tnis the nth event time of N, then n/λ2 ≤ var (Tn) ≤ n/μ2. Some applications of these results are given.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 12 , Issue 4 , October 1998 , pp. 425 - 444
Copyright
Copyright © Cambridge University Press 1998

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