Kaminsky type functional equations and bivariate residual lifetimes distributions

In this paper, we provide generalizations of the functional equations that characterize the lack-of-memory properties: more specifically, we extend the univariate functional equation introduced by Kaminsky (1983, An aging property of the Gompertz survival function and a discrete analog (Tech. Rep.). Department of Mathematical Statistics, University of Umeå, Sweden) and the corresponding bivariate strong and weak versions studied in Marshall and Olkin (2015, A bivariate Gompertz–Makeham life distribution. Journal of Multivariate Analysis, 139, 219–226. https://doi.org/10.1016/j.jmva.2015.02.011) by allowing the conditional survival distribution to be a fully general time dependent distortion of the unconditional one. 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 focuses 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, we study the impact of the time dependent distortion on the aging properties and on the dependence structure of the residual lifetimes via time-varying Kendall's function and tail dependence coefficients: moreover, we provide some illustrative examples showing that these distributions can model both broken hearth effect as well as its reverse version.