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March, 1993 Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators
Sara van de Geer
Ann. Statist. 21(1): 14-44 (March, 1993). DOI: 10.1214/aos/1176349013

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.

Citation

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Sara van de Geer. "Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators." Ann. Statist. 21 (1) 14 - 44, March, 1993. https://doi.org/10.1214/aos/1176349013

Information

Published: March, 1993
First available in Project Euclid: 12 April 2007

zbMATH: 0779.62033
MathSciNet: MR1212164
Digital Object Identifier: 10.1214/aos/1176349013

Subjects:
Primary: 62G05
Secondary: 60G50, 62F12

Rights: Copyright © 1993 Institute of Mathematical Statistics

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Vol.21 • No. 1 • March, 1993
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