Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T08:01:12.460Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bremaud, P. (1981). Point processes and queues: Martingale dynamics. New York: Springer-Verlag.CrossRefGoogle Scholar
2.Brown, M. (1980). Bounds, inequalities and monotonicity properties for some specialized renewal processes. The Annals of Probability 8: 227–240.Google Scholar
3.Brown, M. (1990). On a correlation inequality and its applications. In Block, H.W., Sampson, A.R. & Savits, T.H. (eds.), Topics in statistical dependence. Institute of the Mathematical Statistics Lecture Notes/Monograph Series. 16, pp. 111–119.CrossRefGoogle Scholar
4.Efron, B. & Johnstone, I.M. (1990). Fisher's information in terms of the hazard rate. The Annals of Statistics 18: 38–62.CrossRefGoogle Scholar
5.James, I.R. (1986). On estimating equations with censored data. Biometrika 73: 35–42.Google Scholar
6.Shaked, M. & Shanthikumar, J.G. (1987). The multivariate hazard construction. Stochastic Processes and Their Applications 24: 241–258.CrossRefGoogle Scholar
7.Shaked, M. & Shanthikumar, J.G. (1990). Parametric stochastic convexity and concavity of stochastic processes. Annals of the Institute of Statistical Mathematics 42: 509–531.CrossRefGoogle Scholar
8.Shaked, M. & Shanthikumar, J.G. (1993). Stochastic orders and their applications. New York: Academic Press.Google Scholar
9.Shanthikumar, J.G. & Yao, D.D. (1987). The preservation of likelihood ratio ordering under convolution. Stochastic Processes and Their Applications 23: 259–267.CrossRefGoogle Scholar