On strong unimodality of multivariate discrete distributions

A discrete function f defined on Z(n) is said to be logconcave if f (lambda x+(1-lambda)y) >= [f(x)](lambda)[f(y)](1-lambda) for x, y, lambda x+(1-lambda)y is an element of Z(n). A more restrictive notion is strong unimodality. Following Barndorff-Nielsen [O. Barndorff-Nielsen, Unimodality and exponential families, Commun. Statist. 1 (1973) 189-216] a discrete function p(z), z is an element of Z(n) is called strongly unimodal if there exists a convex function f (x), x is an element of R-n such that f(x) = -log p(x) if x is an element of Z(n). In this paper sufficient conditions that ensure the strong unimodality of a multivariate discrete distribution, are given. Examples of strongly unimodal multivariate discrete distributions are presented. (C) 2008 Elsevier B.V. All rights reserved.