Constrained mixture distributions

Abstract.

The problem of estimating the weights associated with mixture distributions subject to several constraints, such as the percentile and/or moment constraints, is analyzed using the general theory of polyhedral convex cones and systems of inequalities. We address three problems associated with constrained mixture distributions: (a) Compatibility: a set of inequalities is obtained to check whether or not any given set of constraints lead to a feasible solution for the weights, (b) Feasible solutions: a general expression for building feasible solutions for the weights associated with the given constraints is obtained, and (c) Equivalence: the set of all feasible weights is obtained. In addition, the problem is shown to lead to a new mixture distribution, without extra constraints. This new mixture distribution can then be easily used for the statistical analysis (e.g. estimation and hypotheses testing) instead of the original mixture distribution with extra constraints. The proposed methods are illustrated by numerical examples.