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Stochastic ordering of order statistics

Published online by Cambridge University Press:  14 July 2016

A. D. Barbour ,
T. Lindvall  and
L. C. G. Rogers
A. D. Barbour*
Affiliation:
Universität Zürich
T. Lindvall*
Affiliation:
University of Göteborg
L. C. G. Rogers*
Affiliation:
University of Cambridge
*
∗Postal address: Institut für Angewandte Mathematik, Universität Zürich, Rämistrasse 74, CH-8001, Zürich, Switzerland.
∗∗Postal address: Department of Mathematics, University of Göteborg, Sven Hultins gata 6, S-412 96 Göteborg, Sweden.
∗∗∗Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

Keywords

HOMOGENEOUS SAMPLEHETEROGENEOUS SAMPLEEPIDEMIC
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 2 , June 1991 , pp. 278 - 286
Copyright
Copyright © Applied Probability Trust 1991 

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References

Ball, F. (1985) Deterministic and stochastic epidemics with several kinds of susceptibles. Adv. Appl. Prob. 17, 1–22.Google Scholar
Proschan, F. and Sethuraman, J. (1976) Stochastic comparison of order statistics from heterogeneous populations, with applications in reliability. J. Multivariate Anal. 6, 608–616.Google Scholar
Sellke, T. (1983) On the asymptotic distribution of the size of a stochastic epidemic. J. Appl. Prob. 20, 390–394.Google Scholar