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 system. 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 X ˜ t = [ X − t | X > t , Y > s ] 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 X ˜ 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 conditioned inactivity times X ˜ t = [ t − X | X ≤ t , Y ≤ s ] and X t = [ t − X | X ≤ t ] .