A natural parametrization of multivariate distributions with limited memory

作者：Shenkman Natalia^*

来源：Journal of Multivariate Analysis, 2017, 155: 234-251.

DOI：10.1016/j.jmva.2017.01.004

• Lack-of-memory (logarithmically) d-monotone set function
• Wide-sense geometric distribution
• Marshall-Olkin distribution
• Marginal equivalence in minima
• Minimum divisibility
• Multivariate Bernoulli distribution

A unified formulation of the theory of d-variate wide-sense geometric (g) and Marshall-Olkin exponential (Je9d) distributions is presented in which d-monotone set functions occupy a central role. A semi-analytical derivation of gy and dlt(9d 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,y family, and on the marginal equivalence in minima of Cdr' 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.

• 出版日期2017-3