Constrained Nonparametric Maximum Likelihood via Mixtures

Abstract

This article discusses a new technique for calculating maximum likelihood estimators (MLEs) of probability measures when it is assumed the measures are constrained to a compact, convex set. Measures in such sets can be represented as mixtures of simple, known extreme measures, and so the problem of maximizing the likelihood in the constrained measures becomes one of maximizing in an unconstrained mixing measure. Such convex constraints arise in many modeling situations, such as empirical likelihood and estimation under stochastic ordering constraints. This article describes the mixture representation technique for these two situations and presents a data analysis of an experiment in cancer genetics, where a partial stochastic ordering is assumed but the data are incomplete.