A natural parametrization of multivariate distributions with limited memory

Abstract

A unified formulation of the theory of d -variate wide-sense geometric ( G d W ) and Marshall–Olkin exponential ( MO d ) distributions is presented in which d -monotone set functions occupy a central role. A semi-analytical derivation of G d W and MO d distributions is deduced directly from the lack-of-memory property. In this context, the distributions are parametrized with d -monotone and d -log-monotone set functions arising from the univariate marginal distributions of minima and the d -decreasingness of the survival functions. In addition, a one-to-one correspondence is established between d -monotone (resp. d -log-monotone) set functions and d -variate (resp. d -variate min-infinitely divisible) Bernoulli distributions. The advantage of such a parametrization is that it makes the distributions highly tractable. As a showcase, we derive new results on the minimum stability and divisibility of the G d W family, and on the marginal equivalence in minima of G d W and distributions with geometric minima. Similarly, a surprisingly simple proof is given of the prominent result of Esary and Marshall (1974) on the marginal equivalence in minima of multivariate exponential distributions.