Properties of the Maximum Likelihood Estimator of a Mixing Distribution

Abstract

Given a random sample from a mixture density, the objectives is to estimate the mixing distribution by maximum likelihood. Certain properties of this estimator are identified. The estimator is characterized by a family of inequalities which correspond to the usual parametric likelihood equations. If the atomic densities underlying the mixture are of exponential type, it is demonstrated that the estimator matches the first sample moment to the first theoretical moment evaluated at the estimator. However, the sample variance is related to the theoretical variance by an inequality. Following this structural analysis, several algorithms for the estimator are discussed. The paper concludes with an example and a discussion of the difficulties of an asymptotic distribution theory.