Characterizations of multivariate distributions with limited memory revisited: An analytical approach

Abstract

. Alternative proofs of the characterizations of the wide-sense geometric and of the Marshall-Olkin exponential distributions via monotone set functions are provided. In contrast to the ones presented in Shenkman (2017), which rely on the generative constructions of Arnold (1975) or Marshall and Olkin (1967) to establish that certain functions equipped with monotone parameters are proper survival functions, we aim herein to check that these candidates satisfy a set of well known necessary and sufficient analytical conditions. The major difficulty in such an approach consists in verifying that they do not infringe any of the so-called rectangle inequalities. Fortunately, a factorization shows that compliance is guaranteed as long as a finite number of very specific “basis” rectangle inequalities are not violated: a condition which is, by the very definition of the monotone parameters, trivially met.