On the role of dependence in residual lifetimes

Consider a vector (X,Y) that describes the failure times of two non-independent components of a s...ystem. Assuming that the first component has survived up to a given time t>0, i.e., assuming X>t, one can define the corresponding residual lifetime under different assumptions on the failure of the second component, having lifetime Y. In particular, one can observe that the second component has not failed before a time s>0 (maybe different from t), thus the conditioned residual lifetime XX_t = [X-t ! X>t, Y>t] can be considered, or one cannot observe Y, and in this case the conditioned residual lifetime X_t = [X-t | X>t] has to be studied. This note deals with conditions on the survival copula of (X,Y) such that XX_t and X_t are comparable according to the main reliability stochastic orders. Similar conditions, based on the connecting copula of (X,Y), are described also for the corresponding conditioned inactivity times