The Bivariate Lack-of-Memory Distributions

TL;DR

This paper provides a unified framework for bivariate lack-of-memory distributions, deriving new properties, conditions for total positivity, and extending known results like Slepian's inequality.

Contribution

It introduces a comprehensive approach to BLM distributions, establishing new properties and improving existing results, including total positivity and dependence structures.

Findings

Marshall--Olkin survival copula is totally positive of all orders.

Slepian's inequality holds for BLM distributions.

New conditions for total positivity of BLM survival functions.

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.