Stochastic comparisons of conditional residual lifetimes with applications

ABSTRACT

It is not uncommon that the living situation of a unit (e.g. a component in a reliability system) plays a critical role in affecting the performance of another unit or even the whole system. The study on stochastic behaviors of conditional residual lifetimes appears to be particularly important for analyzing the performances of interested units. This paper studies stochastic comparisons on the conditional residual lifetimes of units under the dependent information when another unit has survived up to a given time. Sufficient (and necessary) conditions are established for comparing conditional residual lifetimes of different units under various dependence structures as well as different events according to some traditional stochastic orders. We provide numerical examples based on the well-known FGM copula and Archimedean copulas to verify the conditions. Three real scenarios in reliability theory and risk theory are also considered to show the applicability of our results.

KEYWORDS:

• Conditional residual lifetime
• copula
• stochastic orders
• reliability theory
• risk measure

JEL CLASSIFICATION CODES:

• 90B25
• 60E15

Acknowledgements

The author is grateful to the insightful comments and suggestions from an Editor and two anonymous reviewers, which have improved the presentation of the paper. The author also acknowledges the National Natural Science Foundation of China (No. 12101336).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. Henceforth, the term “dependence structure” stands for the joint (survival) copula shared by $(X,Y)$.

2. We shall relax such kind of assumption in Section 5 to the case where $(X_1,Y)$ and $(X_2,Y)$ may have different copulas. Besides, the copula $\hat{C}$ could be asymmetric in $(u,v)$, that is, $\hat{C}(u,v)\ne \hat{C}(v,u)$, for some $u\ne v$.