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Multivariate dispersive ordering of epoch times of nonhomogeneous Poisson processes

Part of: Special processes Stochastic processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Felíx Belzunce  and
José-María Ruiz
Felíx Belzunce*
Affiliation:
Universidad de Murcia
José-María Ruiz*
Affiliation:
Universidad de Murcia
*
∗ Postal address: Departmento de Estadística e Investigación Operativa, Campus de Espinardo, 30100 Espinardo, Murcia, Spain.
∗ Postal address: Departmento de Estadística e Investigación Operativa, Campus de Espinardo, 30100 Espinardo, Murcia, Spain.
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Abstract

In this paper we find conditions under which the epoch times of two nonhomogeneous Poisson processes are ordered in the multivariate dispersive order. Some consequences and examples of this result are given. These results extend a recent result of Brown and Shanthikumar (1998).

Keywords

Nonhomogeneous Poisson processmultivariate dispersive orderst:icx order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 60G55: Point processes
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 3 , September 2002 , pp. 637 - 643
Copyright
Copyright © Applied Probability Trust 2002 

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References

Ascher, H., and Feingold, H. (1984). Repairable Systems Reliability. Marcel Decker, New York.Google Scholar
Bagai, I., and Kochar, S. C. (1986). On tail ordering and comparison of failure rates. Commun. Statist. Theory Meth. 15, 1377–1388.Google Scholar
Barlow, R. E., and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Bartoszewicz, J. (1985). Dispersive ordering and monotone failure rate distributions. Adv. Appl. Prob. 17, 472–474.Google Scholar
Belzunce, F., and Shaked, M. (2001). Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability. Statist. Prob. Lett. 53, 363–372.Google Scholar
Belzunce, F., Lillo, R., Ruiz, J. M., and Shaked, M. (2001). Stochastic comparisons of nonhomogeneous processes. Prob. Eng. Inf. Sci. 15, 199–224.Google Scholar
Block, H. W., Borges, W. S., and Savits, T. H. (1985). Age-dependent minimal repair. J. Appl. Prob. 22, 370–385.Google Scholar
Brown, M., and Shanthikumar, J. G. (1998). Comparing the variability of random variables and point processes. Prob. Eng. Inf. Sci. 12, 425–444.Google Scholar
Cox, R., and Lewis, P. A. (1966). Statistical Analysis of Series of Events. Methuen, London.CrossRefGoogle Scholar
Crow, L. H. (1974). Reliability analysis for complex, repairable systems. In Reliability and Biometry, eds. Proschan, F. and Serfling, R. J., SIAM, Philadelphia, pp. 379–410.Google Scholar
Gupta, R. C., and Kirmani, S. N. U. A. (1988). Closure and monotonicity properties of nonhomogeneous Poisson processes and record values. Prob. Eng. Inf. Sci. 2, 475–484.Google Scholar
Kuo, L., and Yang, T. Y. (1996). Bayesian computation for nonhomogeneous Poisson processes in software reliability. J. Amer. Statist. Assoc. 91, 763–773.Google Scholar
Lewis, T., and Thompson, J. W. (1981). Dispersive distributions, and the connection between dispersivity and strong unimodality. J. Appl. Prob. 18, 76–90.Google Scholar
Lillo, R. E., Nanda, A. K., and Shaked, M. (2000). Some shifted stochastic orders. In Recent Advances in Reliability Theory, eds. Limnios, N. and Nikulin, M., Birkhäuser, Boston, pp. 85–103.Google Scholar
Musa, J. D., and Okumoto, K. (1984). A logaritmic Poisson execution time model for software reliability measurement. In Proc. 7th Internat. Conf. Software Eng. (26–29 March 1984, Orlando, FL), IEEE Computer Society, Los Alamitos, CA, pp. 230–238.Google Scholar
Rüschendorff, L. (1983). Solution of a statistical optimization problem by rearrangement methods. Metrika 30, 55–61.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, New York.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1998). Two variability orders. Prob. Eng. Inf. Sci. 12, 1–23.CrossRefGoogle Scholar