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Dynamic conditional marginal distributions in reliability theory

Part of: Special 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, AZ 85721, USA.
∗∗ Postal address: Walter A. Haas School of Business, University of California, Berkeley, CA 94720, USA.
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Abstract

Motivated by the need of studying a subset of components, ‘separate' from the other components, we introduce a new definition of ‘marginal distribution'. This is done by fixing the lives of the other components, but without the ‘knowledge' of the components of interest. Formally this is done by minimally repairing the components of no interest up to a predetermined time. Preservation properties of these ‘conditional marginal distributions', with respect to several stochastic orderings, are obtained. Also, inheritance of positive dependence properties, by the conditional marginal distributions, is shown. In addition, the preservations of dynamic multivariate aging properties, by the dynamic conditional marginal distributions, are obtained. The definitions and results are illustrated by a set of examples. Some applications for modelling ‘combinations' of sets of random lifetimes, and for bounding complex sets of random lifetimes, are described.

Keywords

MULTIVARIATE IFRSTOCHASTIC ORDERINGSLOAD-SHARINGFREUND DISTRIBUTIONMULTIVARIATE CONDITIONAL HAZARD RATE FUNCTIONSMAJOR AND MINOR COMPONENTSPOSITIVE DEPENDENCE NOTIONSSTOCHASTIC BOUNDS

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 421 - 428
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research supported by the AFOSR Grant AFOSR-90–0201. Reproduction in whole or in part is permitted for any purpose by the United States Government.

References

Arjas, E. and Norros, I. (1984) Life length and association: A dynamic approach. Math. Operat. Res. 9, 151–158.Google Scholar
Arjas, E. and Norros, I. (1989) Changes of life distribution via a hazard transformation: An inequality with application to minimal repair. Math. Operat. Res. 14, 355–361.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967) Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
Freund, J. E. (1961) A bivariate extension of the exponential distribution. J. Amer. Statist. Assoc. 56, 971–977.Google Scholar
Natvig, B. (1990) On information-based minimal repair and the reduction in remaining system lifetime due to the failure of a specific module. J. Appl. Prob. 27, 365–375.Google Scholar
Norros, I. (1985), Systems weakened by failures. Stoch. Proc. Appl. 20, 181–196.Google Scholar
Ross, S. M. (1984) A model in which component failure rates depend on the working set. Naval Res. Logist. Quart. 31, 297–300.Google Scholar
Schechner, Z. (1984) A load-sharing model: The linear breakdown rule. Naval Res. Logist. Quart. 31, 137–144.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1986) Multivariate imperfect repair. Operat. Res. 34, 437–448.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1990a) Multivariate stochastic orderings and positive dependence in reliability theory. Math. Oper. Res. 15, 545–552.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (1990b) Reliability and maintainability. In Handbook of Operations Research and Management Sciences, eds. Heyman, D. and Sobel, M., Vol. 2, 653–713.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1991a) Dynamic multivariate aging notions in reliability theory. Stoch. Proc. Appl. 38, 85–97.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1991b) Dynamic multivariate mean residual life functions. J. Appl. Prob. 28, 613–629.Google Scholar
Shaked, M. and Shanthikumar, J. G. (1992) Dynamic conditional marginal distributions in reliability theory. Technical Report, Department of Mathematics, University of Arizona, Tucson, AZ 85721.Google Scholar