On the role of dependence in residual lifetimes

Consider (X,Y) describing the failure times of two non-indep. components of a system. Assuming that the first component has survived up to a given time t, X>t, we define the corresponding residual lifetime under different assumptions on the failure of the second component, having lifetime Y. In particular, we observe that the second component has not failed before a time s, (maybe different from t), thus the conditioned residual lifetime X*_t=[X-t| X>t, Y>s] is 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 X*_t and X_t are comparable according to the main stochastic orders.