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Residual Probability Function for Dependent Lifetimes

Author

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  • Mhamed Mesfioui

    (Département de Mathématiques et D’informatique, Université du Québec à Trois-Rivières, 3351 Boulevard des Forges, Trois-Rivières, QC G9A 5H7, Canada)

  • Mohamed Kayid

    (Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia)

Abstract

In this paper, the residual probability function is applied to analyze the survival probability of two used components relative to each other in the case when their lifetimes are dependent. The expression of the function by copulas has been derived along with some examples of particular copulas. The behaviour of the residual probability function in terms of the underlying dependence is also discussed. The residual probability order is also considered in the dependent case. In the class of Archimedean survival copulas, we prove that the residual probability order implies the usual stochastic order in the reversed direction, and the hazard rate order concludes the residual probability order.

Suggested Citation

  • Mhamed Mesfioui & Mohamed Kayid, 2021. "Residual Probability Function for Dependent Lifetimes," Mathematics, MDPI, vol. 9(15), pages 1-13, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:15:p:1782-:d:603076
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    References listed on IDEAS

    as
    1. Maxim Finkelstein & Veronica Esaulova, 2005. "On the weak IFR aging of bivariate lifetime distributions," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 21(3), pages 265-272, May.
    2. Gupta, Nitin & Misra, Neeraj & Kumar, Somesh, 2015. "Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components," European Journal of Operational Research, Elsevier, vol. 240(2), pages 425-430.
    3. M. Kayid & S. Izadkhah & S. Alshami, 2014. "Residual Probability Function, Associated Orderings, and Related Aging Classes," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-10, December.
    4. Bassan, Bruno & Spizzichino, Fabio, 2005. "Bivariate survival models with Clayton aging functions," Insurance: Mathematics and Economics, Elsevier, vol. 37(1), pages 6-12, August.
    5. V. Zardasht & M. Asadi, 2010. "Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(4), pages 460-481, November.
    6. M. Kayid, 2011. "Preservation properties of the moment generating function ordering of residual lives," Statistical Papers, Springer, vol. 52(3), pages 523-529, August.
    7. Sabrina Mulinacci, 2018. "Archimedean-based Marshall-Olkin Distributions and Related Dependence Structures," Methodology and Computing in Applied Probability, Springer, vol. 20(1), pages 205-236, March.
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