The Annals of Statistics

Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators

Sara van de Geer

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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.

Article information

Source
Ann. Statist., Volume 21, Number 1 (1993), 14-44.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349013

Digital Object Identifier
doi:10.1214/aos/1176349013

Mathematical Reviews number (MathSciNet)
MR1212164

Zentralblatt MATH identifier
0779.62033

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 60G50: Sums of independent random variables; random walks 62F12: Asymptotic properties of estimators

Keywords
Consistency empirical process entropy Hellinger distance maximum likelihood rates of convergence

Citation

van de Geer, Sara. Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators. Ann. Statist. 21 (1993), no. 1, 14--44. doi:10.1214/aos/1176349013. https://projecteuclid.org/euclid.aos/1176349013


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