Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T21:32:49.585Z Has data issue: false hasContentIssue false
  • English
  • Français

A stochastic ordering of partial sums of independent random variables and of some random processes

Part of: Distribution theory - Probability Survival analysis and censored data

Published online by Cambridge University Press:  14 July 2016

Philip J. Boland ,
Frank Proschan  and
Y. L. Tong
Philip J. Boland*
Affiliation:
University College, Dublin
Frank Proschan*
Affiliation:
Florida State University
Y. L. Tong*
Affiliation:
Georgia Institute of Technology
*
∗Postal address: Department of Statistics, University College Dublin, Belfield, Dublin 4, Ireland.
∗∗Postal address: Department of Statistics, Florida State University, Tallahassee, FL 32306, USA.
∗∗∗Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

Keywords

ARRANGEMENT-DECREASING FUNCTIONSPOISSON PROCESSWIENER PROCESS

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62N05: Reliability and life testing
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 645 - 654
Copyright
Copyright © Applied Probability Trust 1992 

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.)

Footnotes

Research partially supported by AFOSR under Grant Nos 88–0040 and 89–0221.

Research partially supported by AFOSR under Grant Nos 88–0040 and 89–0221.

Research partially supported by the NSF under Grant No DMS-9001721.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart, and Winston, New York.Google Scholar
Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.Google Scholar
Marshall, A. W. and Olkin, I. (1979) Inequalities: Theory of Majorization and Its Applications. Academic Press, New York.Google Scholar
Pledger, G. and Proschan, F. (1973) Stochastic comparisons of random processes, with applications in reliability. J. Appl. Prob. 10, 572–585.Google Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Shaked, M. and Shanthikumar, G. (1987) The multivariate hazard construction. Stoch. Proc. Appl. 24, 241–258.Google Scholar
Veinott, A. F. (1965) Optimal policy in a dynamic, single product, nonstationary inventory model with several demand classes. Operat. Res. 13, 761–778.Google Scholar
Wald, A. (1947) Sequential Analysis. Wiley, New York.Google Scholar