Generalized bivariate strong and weak lack-of-memory properties: characterization and dependence structure

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It is well known that many distributions used in survival analysis do not satisfy standard lack-of-memory properties. In this article, we consider the following functional equations F¯(s1+t1,s2+t2)=F¯(s1,s2)⊗hF¯(t1,t2), ∀ t1,t2,s1,s2≥0 and F¯(s1+t,s2+t)=F¯(s1,s2)⊗hF¯(t,t), ∀ t,s1,s2≥0, where a⊗hb=h(h−1(a)⋅h−1(b)), ∀ a,b∈[0,1] with h an increasing bijection of [0,1], called generator. We call the former “pseudo strong functional equation” and the latter “pseudo weak functional equation”, that generalize the functional equations that characterize the bivariate standard strong and weak lack-of-memory properties, recovered when h(x)=id. After finding sufficient and necessary conditions under which the solutions of the above functional equations are bivariate survival functions, we focus our attention on the distributions which satisfy the pseudo weak functional equation. In particular, we provide a characterization, in line with the existing literature, of these distributions and we investigate their dependence structure, analyzing the impact that the generator h has on their Kendall distribution functions and on their tail dependence coefficients.