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Characterization of Some First Passage Times Using Log-Concavity and Log-Convexity as Aging Notions

Published online by Cambridge University Press:  27 July 2009

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked
Affiliation:
Department of MathematicsUniversity of Arizona, Tucson, Arizona 85721
J. George Shanthikumar
Affiliation:
School of Business AdministrationUniversity of California, BerkeleyBerkeley, California 94720
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Abstract

An interpretation of log-concavity and log-convexity as aging notions is given in this paper. It imitates a stochastic ordering characterization of the NBU (new better than used) and the NWU (new worse than used) notions but stochastic ordering is now replaced by the likelihood ratio ordering. The new characterization of log-concavity and log-convexity sheds new light on these properties and enables one to obtain intuitively simple proofs of the log-convexity and log-concavity of some first passage times of interest in branching processes and in reliability theory.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 1 , Issue 3 , July 1987 , pp. 279 - 291
Copyright
Copyright © Cambridge University Press 1987

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References

1.Assaf, D., Shaked, M., & Shanthikumar, J. G. (1985). First passage times with PFr densities. J. Appl. Probab. 22: 185–196.Google Scholar
2.Barlow, R. E., & Proschan, F. (1975). Statistical theory of reliability and life testing: Probability models. New York: Holt, Rinehart, and Winston.Google Scholar
3.Bhattacharjee, M. C. (1986). Reliability ideas and applications in economics and social science. In Krishnaiah, P. R. (ed.), Development in statistics: Reliability and quality control (vol. 7). New York: Academic Press.Google Scholar
4.Bhattacharjee, M. C. (1986). The time to extinction of branching processes and log-convexity: I. Prob. in the Eng. and Inf. Sciences, 1:265–278.CrossRefGoogle Scholar
5.Brown, M., & Chaganty, N. R. (1983). On the first passage time distribution for a class of Markov chains. Ann. Probab. 11: 1000–1008.CrossRefGoogle Scholar
6.Derman, C., Ross, S. M., & Schechner, Z. (1983). A note on first passage times in birth and death and nonnegative diffusion processes. Naval Res. Logist. Quart. 30: 283–285.Google Scholar
7.Harris, T. E. (1963). The theory of branching processes. Berlin: Springer Verlag.CrossRefGoogle Scholar
8.Karlin, S. (1964). Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc. 111: 33–107.CrossRefGoogle Scholar
9.Karlin, S. (1968). Total positivity, Vol. I. Stanford, California: Stanford University Press.Google Scholar
10.Keilson, J., & Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10 181–198.Google Scholar
11.Klefsjo, B. (1984). Reliability interpretations of some concepts from economics. Naval Res. Logist. Quart. 31: 301–308.Google Scholar
12.Ross, S. M. (1983). Stochastic processes. New York: Wiley.Google Scholar
13.Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: Wiley.Google Scholar
14.Whitt, W. (1980). Uniform conditional stochastic order. J. Appl. Probab. 17: 112–123.Google Scholar