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Stochastic models for dependent life lengths induced by common pure jump shock environments

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Haijun Li
Haijun Li*
Affiliation:
Washington State University
*
∗Postal address: Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99164, USA. Email address: lih@haijun.math.wsu.edu
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Abstract

The lifelengths of components of a system are usually dependent due to the common random production and operating environments. In this paper, we introduce a multi-variate pure jump Markov process to describe a large class of damage processes on various system components driven by common environmental shocks, and establish some dependence properties (association) for such a process and its multivariate increment process. These strong association properties describe both spatial dependence and temporal dependence of a multivariate pure jump process, and also provide a vehicle to derive some structural properties of component lifelengths of the systems operating in such an environment. Some bounds for the joint survival functions of component lifelengths are also obtained.

Keywords

Survival in random environmentsmultivariate pure jump processhazard rate processassociation of probability measures on partially ordered spacesassociation in timestochastic integral equation

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 453 - 469
Copyright
Copyright © by the Applied Probability Trust 2000 

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Footnotes

Supported in part by the NSF grant DMI 9812994.

References

Barlow, R. E., and Proschan, F. (1981). Statistical Theory of Reliability and Life Testing. To Begin With, Silver Spring, MD.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Brown, T. C., and Donnelly, P. (1993). On the conditional intensities and on interparticle correlation in non-linear death processes. Adv. Appl. Prob. 25, 255–260.Google Scholar
Ҫinlar, E., and Jacod, J. (1981). Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes, eds Ҫinlar, E., Chung, K. L. and Getoor, R. K. Birkhäuser, Boston, MA.Google Scholar
Ҫinlar, E. and Özekici, S. (1987). Reliability of complex devices in random environments. Prob. Eng. Inf. Sci. 1, 97–115.Google Scholar
Drosen, J. W. (1986). Pure jump shock models in reliability. Adv. Appl. Prob. 18, 423–440.Google Scholar
Esary, J. D., and Proschan, F. (1970). A reliability bound for systems of maintained, interdependent components. J. Amer. Statist. Assoc. 65, 329–338.Google Scholar
Esary, J. D., Proschan, F., and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.CrossRefGoogle Scholar
Glasserman, P. (1992). Processes with associated increments, J. Appl. Prob. 29, 313–333.Google Scholar
Joag-Dev, K., and Proschan, F. (1983). Negative association of random variables with applications. Ann. Statis. 11, 286–295.Google Scholar
Jogdeo, K. (1978). On a probability bound of Marshall and Olkin. Ann. Statist. 6, 232–234.Google Scholar
Kamae, T., Krengel, U., and O'Brien, G. L. (1977). Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.Google Scholar
Kebir, Y. (1991). On the hazard rate processes. Naval Res. Logist. 38, 865–876.Google Scholar
Kijima, M., Li, H., and Shaked, M. (1998). Stochastic processes in reliability. To appear in Hanbook of Statistics 18, eds Rao, C. R. and Shanbhag, D. N. Preprint.Google Scholar
Lefèvre, C., and Milhaud, X. (1990). On the association of the lifelengths of components subjected to a stochastic environment. Adv. Appl. Prob. 22, 961–964.Google Scholar
Li, H., and Shaked, M. (1995). On the first passage times for Markov processes with monotone convex transition kernels. Stoch. Proc. Appl. 58, 205–216.Google Scholar
Li, H., and Shaked, M. (1997). Aging first-passage times of Markov processes: a matrix approach. J. Appl. Prob. 34, 1–13.Google Scholar
Lindqvist, H. (1987). Monotone and associated Markov chains, with applications to reliability theory. J. Appl. Prob. 24, 679–695.Google Scholar
Lindqvist, H. (1988). Association of probability measures on partially ordered spaces. J. Multivariate Anal. 26, 111–132.Google Scholar
Shaked, M., and Shanthikumar, J. G. (1988). On the first-passage times of pure jump processes. J. Appl. Prob 25, 501–509.Google Scholar
Singpurwalla, N. (1995). Survival in dynamic environments. Statis. Sci. 10, 86–103.Google Scholar
Stone, C. (1963). Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14, 694–697.Google Scholar
Tong, Y. L. (1980). Probability Inequalities in Multivariate Distributions. Academic Press, New York.Google Scholar