Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators

Abstract

Consider a class $\\mathscr{P}={P_\\theta:\\theta\\in\\Theta}$ of probability measures on a measurable space $(\\mathscr{X},\\mathscr{A})$, dominated by a $\\sigma$ -finite measure $\\mu$. Let $f_\\theta=dP_\\theta/d_\\mu$, $\\theta\\ in\\Theta$, and let $\\theta_n$ be a maximum likelihood estimator based on n independent observations from $P_{\\theta_0}$, $\\theta_0\\in\\Theta$. We use results from empirical process theory to obtain convergence for the Hellinger distance $h(f_{\\hat{\\theta}_n}, f_{\\theta_0})$, under certain entropy conditions on the class of densities ${f_\\theta:\\theta\\in\\Theta}$ The examples we present are a model with interval censored observations, smooth densities, monotone densities and convolution models. In most examples, the convexity of the class of densities is of special importance.