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Order-Preserving Shock Models

Published online by Cambridge University Press:  27 July 2009

Y. Kebir
Y. Kebir
Affiliation:
Departments of Management Science and Mathematical Sciences, Loyola University of Chicago, Chicago, Illinois 60611
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Abstract

To date, research in shock models has been primarily concerned with the classpreserving properties of certain shock and damage kernels. This article focuses on the order-preserving properties of those kernels. We show that they preserve stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 8 , Issue 1 , January 1994 , pp. 125 - 134
Copyright
Copyright © Cambridge University Press 1994

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References

1.A-Hameed, M.S. & Proschan, F. (1973). Non-stationary shock models. Stochastic Processes and Their Applications 1: 383–404.CrossRefGoogle Scholar
2.A-Hameed, M.S. & Proschan, F. (1975). Shock models with underlying birth process. Journal of Applied Probability 12: 18–28.CrossRefGoogle Scholar
3.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, MD: To Begin With.Google Scholar
4.Block, H.W. & Savits, T. (1978). Shock models with NBUE survival. Journal of Applied Probability 15: 621–628.CrossRefGoogle Scholar
5.Çinlar, E. (1975). Introduction to stochastic processes. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
6.Deshpande, J.V. et al. , (1990). Some partial orderings describing positive ageing. Communications in Statistics — Stochastic Models 6: 471–481.CrossRefGoogle Scholar
7.Esary, J.D., Marshall, A.W., & Proschan, F. (1973). Shock models and wear processes. Annals of Probability 1: 627–649.CrossRefGoogle Scholar
8.Ghosh, M. & Ebrahimi, N. (1982). Shock models leading to increasing failure rate and decreasing mean residual life survival. Journal of Applied Probability 19: 158–166.CrossRefGoogle Scholar
9.Karlin, S. (1968). Total positivity. Vol. I. Stanford, CA: Stanford University Press.Google Scholar
10.Karlin, S. & Proschan, F. (1960). Polya type distributions of convolutions. Annals of Mathematical Statistics 31: 721–736.CrossRefGoogle Scholar
11.Keilson, J. & Sumita, U. (1982). Uniform stochastic ordering and related inequalities. Canadian Journal of Statistics 10: 181–198.CrossRefGoogle Scholar
12.Klefsjo, B. (1981). HNBUE survival under some shock models. Scandinavian Journal of Statistics 8: 39–47.Google Scholar
13.Kochar, S.C. (1990). On preservation of some partial orderings under shock models. Advances in Applied Probability 22: 508–509.CrossRefGoogle Scholar
14.Lynch, J., Mimmack, G., & Proschan, F. (1987). Uniform stochastic ordering and total positivity. Canadian Journal of Statistics 13: 63–69.CrossRefGoogle Scholar
15.Pellerey, F. (1993). Partial orderings under cumulative shock models. Advances in Applied Probability 4 (to appear).CrossRefGoogle Scholar
16.Ross, S.M. (1983). Stochastic processes. New York: John Wiley.Google Scholar
17.Singh, H. & Jain, K. (1989). Preservation of some partial orderings under Poisson shock models. Advances in Applied Probability 21: 713–716.CrossRefGoogle Scholar
18.Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: John Wiley.Google Scholar