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) and is said to be reducible if x*y=x*z if and only if y=z and if y*w=z*w if and only if y=z. The operation is said to have an identity element e if x*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].