The Bivariate Lack-of-Memory Distributions

Abstract

We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall--Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian's inequality also holds true for BLM distributions

An Identity for Two Integral Transforms Applied to the Uniqueness of a Distribution via its Laplace-Stieltjes Transform

“…where θ is a positive constant (see Marshall andOlkin 1967, or Barlow andProschan 1981, p. 130). For convenience, we denote H = BLM(F, G, θ), which has a singular part on the line x = y with probability p(θ) := (f (0) + g(0))/θ − 1 ≥ 0, where f (0) = lim ε→0 + F (ε)/ε, and g(0) = lim ε→0 + G(ε)/ε (see Remark 2 in Lin et al 2019).…”

“…where θ > 0 is a constant, if and only if H is the bivariate lack-of-memory distribution BLM(F, G, θ) with survival function (see Section 5 or Lin et al 2019):…”