A Generalization of Bivariate Lack-of-Memory Properties

Abstract

In this paper, we propose an extension of the standard strong and weak lack-of-memory properties. We say that the survival function $\bar{F}$ of the vector $(X,Y)$ satisfies pseudo lack-of-memory property in strong version if \begin{equation} \label{strong}\bar F_{X,Y}(s_1+t_1,s_2+t_2)=\bar F_{X,Y}(s_1,s_2)\otimes_h\bar F_{X,Y}(t_1,t_2), \ t_1,t_2,s_1,s_2 \geq 0 \end{equation} and in weak version if \begin{equation}\label{weak}\bar F_{X,Y}(s_1+t,s_2+t)=\bar F_{X,Y}(s_1,s_2)\otimes_h\bar F_{X,Y}(t,t), \ s_1,s_2,t \geq 0\end{equation} with $a\otimes_hb=h\left (h^{-1}(a)\cdot h^{-1}(b)\right )$, where $h$ is an increasing bijection of $[0,1]$, called generator. After finding sufficient conditions under which the solutions of the above functional equations are bivariate survival functions, we focus on distributions satisfying the latter: we study specific properties in comparison with standard lack-of-memory property and we give a characterization in terms of the random variables $\min(X,Y)$ and $ X -Y$. Finally, we investigate the induced dependence structure, determining their singularity in full generality and studying the upper and lower dependence coefficients for some specific choices of the marginal survival functions and of the generator $h$.