A natural parametrization of multivariate distributions with limited memory

A unified formulation of the theory of d-variate wide-sense geometric (GdW) and Marshall-Olkin exponential (MOd) distributions is presented in which d-monotone set functions occupy a central role. A semi-analytical derivation of GdW and MOd 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 GdW family, and on the marginal equivalence in minima of GdW 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.