Stochastic comparisons of conditional residual lifetimes with applications

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.