Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T05:37:37.598Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterization of life distributions under some generalized stochastic orderings

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

Published online by Cambridge University Press:  14 July 2016

Jun Cai  and
Yanhong Wu
Jun Cai*
Affiliation:
Concordia University
Yanhong Wu*
Affiliation:
University of Alberta
*
∗Postal address: Department of Mathematics and Statistics, Concordia University, Montreal, Quebec, H4B 1R6, Canada.
∗∗Postal address: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, T6G 2G1, Canada.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the characterizations of life distributions under four stochastic orderings, < p, < (p), < (p) and < L, by a unified method. Conditions for the stochastic equality of two non-negative random variables under the four stochastic orderings are derived. Many previous results are consequences. As applications, we provide characterizations of life distributions by a single value of their Laplace transforms under orderings < p and < (p) and their moment generating functions under orderings < p and < (p). Under ordering < L, a characterization is given by the expected value of a strictly completely monotone function. The conditions for the stochastic equality of two non-negative vectors under the stochastic orderings < (p), < (p) and < L are presented in terms of the Laplace transforms and moment generating functions of their extremes and sample means. Characterizations of the exponential distribution among L and L life distribution classes are also given.

Keywords

CONCAVITYMOMENT GENERATING FUNCTIONSLAPLACE TRANSFORMSTOCHASTIC EQUALITYSTRICTLY COMPLETELY MONOTONE

MSC classification

Primary: 62N05: Reliability and life testing
Secondary: 60E10: Characteristic functions; other transforms
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 3 , September 1997 , pp. 711 - 719
Copyright
Copyright © Applied Probability Trust 1997 

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

Alzaid, A. A. Kim, J. S. and Proschan, F. (1991) Laplace ordering and its applications. J. Appl. Prob. 28, 116–130.CrossRefGoogle Scholar
Baccelli, F. and Makowski, A. M. (1989) Multidimensional stochastic ordering and associated random variables. Operat. Res. 37, 478–487.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1981) Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, CA.Google Scholar
Basu, S. K. and Kirmani, S. N. U. A. (1986) Some results involving HNBUE distributions. J. Appl. Prob. 23, 1038–1044.CrossRefGoogle Scholar
Bhattacharjee, M. C. (1991) Some generalized variability orderings among life distributions with reliability applications. J. Appl. Prob. 28, 374–383.CrossRefGoogle Scholar
Bhattacharjee, M. C. and Sethuraman, J. (1990) Families of life distributions characterizations by two moments. J. Appl. Prob. 27, 720–725.CrossRefGoogle Scholar
Cai, J. (1994) Characterizations of life distributions by moments of extremes and sample mean. J. Appl. Prob. 31, 148–155.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications. Vol. II. 2nd edn. Wiley, New York.Google Scholar
Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distribution. Naval. Res. Logist. Quart. 29, 331–344.CrossRefGoogle Scholar
Klefsjö, B. (1983) A useful ageing property based on the Laplace transform. J. Appl. Prob. 20, 615–626.CrossRefGoogle Scholar
Li, H. and Zhu, H. (1994) Stochastic equivalence of ordered random variables with applications in reliability theory. Statist. Prob. Lett. 20, 384–393.CrossRefGoogle Scholar
Lin, G. D. (1994) On equivalence of weak convergence and moment convergence of life distributions. Ann. Inst. Statist. Math. 46, 587–592.CrossRefGoogle Scholar
Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Scarsini, M. and Shaked, M. (1990) Some conditions for stochastic equality. Naval. Logist. Res. 37, 617–625.3.0.CO;2-L>CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994) Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Stoyan, D. (1983) Comparison Methods for Queues and Other Stochastic Models. Wiley, New York.Google Scholar