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Stochastic Comparisons for Multivariate Shock Models

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  • Pellerey, Franco

Abstract

Consider two devices subjected to shocks arriving according to two identically defined counting processes. Let N1 and N2 be the random numbers of shocks until failure of the two devices, respectively, and let T1 and T2 be their random lifetimes. Conditions such that stochastic orders between N1 and N2 are preserved by T1 and T2 have been investigated in recent literature. Here we study this problem in a multivariate setting, considering systems of non-independent components, and we extend some known results to multivariate stochastic orders. Two kinds of multivariate generalizations are considered; the case that each one of the components is subjected to its own fond of shocks and the case that all the components of the same system are subjected to a common font of shocks.

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  • Pellerey, Franco, 1999. "Stochastic Comparisons for Multivariate Shock Models," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 42-55, October.
    • Handle: RePEc:eee:jmvana:v:71:y:1999:i:1:p:42-55
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    References listed on IDEAS

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    1. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities II. Multivariate reverse rule distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 499-516, December.
    2. Karlin, Samuel & Rinott, Yosef, 1980. "Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 467-498, December.
    3. Savits, Thomas H. & Shaked, Moshe, 1981. "Shock models and the MIFRA property," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 273-283, August.
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    Cited by:

    1. Fierro, Raúl & Leiva, Víctor & Maidana, Jean Paul, 2018. "Cumulative damage and times of occurrence for a multicomponent system: A discrete time approach," Journal of Multivariate Analysis, Elsevier, vol. 168(C), pages 323-333.
    2. J. M. Fernández-Ponce & M. R. Rodríguez-Griñolo, 2017. "New properties of the orthant convex-type stochastic orders," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 618-637, September.
    3. Shaked, Moshe, 2007. "Stochastic comparisons of multivariate random sums in the Laplace transform order, with applications," Statistics & Probability Letters, Elsevier, vol. 77(12), pages 1339-1344, July.
    4. Badía, F.G. & Sangüesa, C. & Cha, J.H., 2014. "Stochastic comparison of multivariate conditionally dependent mixtures," Journal of Multivariate Analysis, Elsevier, vol. 129(C), pages 82-94.
    5. Mesfioui, Mhamed & Denuit, Michel M., 2015. "Comonotonicity, orthant convex order and sums of random variables," Statistics & Probability Letters, Elsevier, vol. 96(C), pages 356-364.
    6. Haijun Li & Susan H. Xu, 2001. "Directionally Convex Comparison of Correlated First Passage Times," Methodology and Computing in Applied Probability, Springer, vol. 3(4), pages 365-378, December.
    7. Kulik, Rafal & Szekli, Ryszard, 2005. "Dependence orderings for some functionals of multivariate point processes," Journal of Multivariate Analysis, Elsevier, vol. 92(1), pages 145-173, January.
    8. Mesfioui, Mhamed & Denuit, Michel, 2014. "Comonotonicity, orthant convex order and sums of random variables," LIDAM Discussion Papers ISBA 2014002, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    9. Hu, Taizhong & Khaledi, Baha-Eldin & Shaked, Moshe, 2003. "Multivariate hazard rate orders," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 173-189, January.

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