Kaminsky Type Functional Equations and Bivariate Residual Lifetimes Distributions

Abstract

This paper considers generalizations of the functional equations that characterize the lack-of-memory properties at univariate and bivariate levels. Specifically, we extend the univariate functional equation introduced by Kaminsky (1983) (that characterizes the Gompertz distribution) and the corresponding bivariate strong and weak versions later studied in Marshall and Olkin (2015) by allowing the conditional survival distribution to be a fully general time dependent distortion of the unconditional one: in particular, we show that the solutions of these generalized functional equations coincide with the solutions of the functional equations studied in Ricci (2024). Since the univariate functional equation leads only to a trivial case and the solutions of the strong bivariate functional equation have been already studied in the literature, the analysis is focused on the weak bivariate case, where joint residual lifetimes are conditioned on survival beyond a common threshold t. In view of potential applications to insurance risk analysis, the impact of the time dependent distortion on the aging properties of the associated distribution is analized as well as the time dependent dependence structure of the residual lifetimes through time-varying versions of the Kendall's function and of the tail dependence coefficients. Many examples are provided and a wide family of bivariate survival distributions satisfying the generalized weak functional equation is constructed through a mixing approach.