Multivariate Positive Dependence on the Simplex

The Liouville and Generalized Liouville families have been proposed as parametric models for data constrained to the simplex. These families have generated practical interest owing primarily to inadequacies, such as a completely negative covariance structure, that are inherent in the better-known Dirichlet class. Although there is some numerical evidence suggesting that the Liouville and Generalized Liouville families can produce completely positive and mixed covariance structures, no general paradigms have been developed. Research toward this end might naturally be focused on the many classical "positive dependence" concepts available in the literature, all of which imply a nonnegative covariance structure. However, in this article it is shown that no strictly positive distribution on the simplex can possess any of these classical dependence properties. The same result holds for Liouville and generalized Liouville distributions even if the condition of strict positivity is relaxed.