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On the first-passage times of pure jump processes

Published online by Cambridge University Press:  14 July 2016

Moshe Shaked  and
J. George Shanthikumar
Moshe Shaked*
Affiliation:
University of Arizona
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Mathematics, University of Arizona, Tucson, AZ85721, USA.
∗∗ Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
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Abstract

Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.

Keywords

IFRIFRALOGCONCAVITYTOTAL POSITIVITYGAMMA PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 3 , September 1988 , pp. 501 - 509
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Supported by U.S. Air Force Office of Scientific Research Grant AFOSR-84-0205.

References

Abdel-Hameed, M. (1984) Life distribution properties of devices subject to a pure jump damage process. J. Appl. Prob. 21, 816–825.Google Scholar
Assaf, D., Shaked, M. and Shanthikumar, J. G. (1985) First-passage times with PFr densities. J. Appl. Prob. 22, 185–196.Google Scholar
Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.Google Scholar
Cinlar, E. (1980) On a generalization of gamma processes. J. Appl. Prob. 17, 467–480.Google Scholar
Cinlar, E. and Jacod, J. (1981) Representation of semimartingales Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes 1981, ed. Cinlar, E., Chung, K. L. and Getoor, R. K., Birkhauser, Boston, 159–242.Google Scholar
Drosen, J. W. (1986) Pure jump shock models in reliability. Adv. Appl. Prob. 18, 423–440.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Griffith, W. S. and Srinivasan, C. (1985) A note on survival models under a Markov process. J. Appl. Prob. 22, 982–988.Google Scholar
Karlin, S. (1964) Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc. 111, 33–107.Google Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford.Google Scholar
Keilson, J. and Sumita, U. (1982) Uniform stochastic ordering and related inequalities. Canad. J. Statist. 10, 181–198.Google Scholar
Shaked, M. and Shanthirumar, J. G. (1987) IFRA properties of some Markov jump processes with general state space. Math. Operat. Res. 12, 562–568.Google Scholar
Shanthirumar, J. G. (1988) DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.Google Scholar
Whitt, W. (1980) Uniform conditional stochastic order, J. Appl. Prob. 17, 112–123.Google Scholar