Risk Bounds for Mixture Density Estimation

Abstract

In this paper we focus on the problem of estimating a boundeddensity using a finite combination of densities from a givenclass. We consider the Maximum Likelihood Procedure (MLE) and the greedy procedure described by Li and Barron. Approximation and estimation bounds are given for the above methods. We extend and improve upon the estimation results of Li and Barron, and in particular prove an $O(\frac{1}{\sqrt{n}})$ bound on the estimation error which does not depend on the number of densities in the estimated combination.