Characterizations using the generalized reversed lack of memory property

A binary operator ∗∗ over real numbers is said to be associative if (x∗y)∗z=x∗(y∗z)(x∗y)∗z=x∗(y∗z) and is said to be reducible if x∗y=x∗zx∗y=x∗z if and only if y=zy=z and if y∗w=z∗wy∗w=z∗w if and only if y=zy=z. The operation is said to have an identity element ee if x∗e=xx∗e=x. In this paper a characterization of a subclass of the reversed generalized Pareto distribution [Castillo, E. Hadi, A.S., 1995. Modelling life time data with applications to fatigue models. Journal of American Statistical Association. 90 (431), 1041–1054] is generalized using this operator. The idea is extended to the bivariate case too and it is shown that it characterizes a class of bivariate distributions containing the characterized extension (CE) model of Roy [Roy, D. 2002a. A characterization of model approach for generating bivariate life distributions using reversed hazard rates. Journal of Japan Statistical Society 32 (2), 239–245].