Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-23T14:45:58.775Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic orders based on ratios of Laplace transforms

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Moshe Shaked  and
Tityik Wong
Moshe Shaked*
Affiliation:
University of Arizona
Tityik Wong*
Affiliation:
Community College of Southern Nevada
*
∗Postal address: Department of Mathematics, Building #89, University of Arizona, Tucson, Arizona 85721, USA.
∗∗Postal address: Department of Mathematics, Community College of Southern Nevada, 3200 E. Cheyenne Ave-SIA, North Las Vegas, Nevada 89030, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to study two notions of stochastic comparisons of non-negative random variables via ratios that are determined by their Laplace transforms. Some interpretations of the new orders are given, and various properties of them are derived. The relationships to other stochastic orders are also studied. Finally, some applications in reliability theory are described.

Keywords

PRESERVATION PROPERTIESINCREASING CONVEX ORDERINCREASING CONCAVE ORDERHAZARD RATE ORDERREVERSE HAZARD RATE ORDERLIKELIHOOD RATIO ORDERUSUAL STOCHASTIC ORDERLAPLACE TRANSFORMLAPLACE TRANSFORM ORDERRANDOM MINIMA AND MAXIMARELIABILITY THEORY

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997 , pp. 404 - 419
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.)

Footnotes

Supported by NSF Grant DMS 9303891.

References

Ahsanullah, M. (1988) Characteristic properties of order statistics based on random sample size from an exponential distribution. Statist. Neerlandica 42, 193–197.CrossRefGoogle Scholar
Alzaid, A., Kim, J. S. and Proschan, F. (1991) Laplace ordering and its applications. J Appl. Prob. 28, 116–130.Google Scholar
Buhrman, J. M. (1973) On order statistics when the sample size has a binomial distribution. Statist. Neerlandica 27, 125–126.Google Scholar
Cohen, J. W. (1974) Some ideas and models in reliability theory. Statist. Neerlandica 28, 1–10.CrossRefGoogle Scholar
Consul, P. C. (1984) On the distributions of order statistics for a random sample size. Statist. Neerlandica 38, 249–256.CrossRefGoogle Scholar
Fagiuoli, E. and Pellerey, F. (1994) Preservation of certain classes of life distributions under Poisson shock models. J. Appl. Prob. 31, 458–465.Google Scholar
Gupta, D. and Gupta, R. C. (1984) On the distribution of order statistics for a random sample size. Statist. Neerlandica 38, 13–19.Google Scholar
Joag-Dev, K., Kochar, S. and Proschan, F. (1995) A general composition theorem and its applications to certain partial orderings of distributions. Statist. Prob. Lett. 22, 111–119.Google Scholar
Karlin, S. (1968) Total Positivity. Vol. I. Stanford University Press, Stanford, CA.Google Scholar
Kim, J. S. and Proschan, F. (1988) Laplace ordering and its reliability applications. Technical report. Department of Statistics, Florida State University.Google Scholar
Kumar, A. (1986) On sampling distributions of order statistics for a random sample size. Acta Ciencia Indica XII, 217–233.Google Scholar
Pellerey, F. (1993) Partial orderings under cumulative damage shock models. Adv. Appl. Prob. 25, 939–946.Google Scholar
Raghunandanan, K. and Patil, S. A. (1972) On order statistics for random sample size. Statist. Neerlandica 26, 121–126.Google Scholar
Rohatgi, V. K. (1987) Distribution of order statistics with random sample size. Commun. Statist. – Theory Meth. 16, 3739–3743.Google Scholar
Shaked, M. (1975) On the distribution of the minimum and of the maximum of a random number of i.i.d. random variables. In Statistical Distributions in Scientific Work. Vol. I. ed. Patil, G. P., Kotz, S. and Ord, J. K. Reidel, Dordrecht. pp. 363–380.Google Scholar
Shaked, M. and Shanthirumar, J. G. (1994) Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Shaked, M. and Wong, T. (1995) Preservation of stochastic orderings under random mapping by point processes. Prob. Eng. Inf. Sci. 9, 563–580.CrossRefGoogle Scholar