Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T09:05:53.155Z Has data issue: false hasContentIssue false
  • English
  • Français

On increasing-failure-rate random variables

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Sheldon M. Ross ,
J. George Shanthikumar  and
Zegang Zhu
Sheldon M. Ross*
Affiliation:
University of Southern California
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
Zegang Zhu*
Affiliation:
University of California, Berkeley
*
∗Postal address: Daniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, 3715 McClintock Avenue, GER 240, Los Angeles, CA 90089-0193, USA. Email address: [email protected]
∗∗Postal address: Department of Industrial Engineering and Operations Research, 4135 Etcheverry Hall, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: Department of Industrial Engineering and Operations Research, 4135 Etcheverry Hall, University of California, Berkeley, CA 94720, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

Keywords

Stochastic orderingIFR propertyDRHR propertyrenewal processMarkov chain

MSC classification

Primary: 62E10: Characterization and structure theory
Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 3 , September 2005 , pp. 797 - 809
Copyright
© Applied Probability Trust 2005 

References

Barlow, R. E. and Marshall, A. W. (1964). Bounds for distributions with monotone hazard rate. Ann. Statist. 36, 1234–1274.Google Scholar
Barlow, R. E. and Marshall, A. W. (1965). Tables of bounds for distributions with monotone hazard rate. J. Amer. Statist. Assoc. 60, 872–890.CrossRefGoogle Scholar
Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Kijima, M. (1989). Uniform monotonicity of Markov processes and its related properties. J. Operat. Res. Soc. Japan 32, 475–490.Google Scholar
Kijima, M. (1992). Further monotonicity properties of renewal processes. Adv. Appl. Prob. 25, 575–588.Google Scholar
Kochar, S. C. (1990). Some partial ordering results on record values. Commun. Statist. Theory Meth. 19, 299–306.CrossRefGoogle Scholar
Li, H. and Shaked, M. (1997). Ageing first-passage times of Markov processes: a matrix approach. J. Appl. Prob. 34, 1–13.CrossRefGoogle Scholar
Pellerey, F., Shaked, M. and Zinn, J. (2000). Nonhomogeneous Poisson processes and logconcavity. Prob. Eng. Inf. Sci. 14, 353–373.Google Scholar
Ross, S. M. (2000). Introduction to Probability Models, 7th edn. Harcourt, Burlington, MA.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1988). On the first-passage times of pure Jump processes. J. Appl. Prob. 25, 501–509.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
Shanthikumar, J. G. (1988). DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.Google Scholar