Mean Residual Lifetime Frailty Models: A Weighted Perspective

The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations' lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.