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February 2010 Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density
Judith Rousseau
Ann. Statist. 38(1): 146-180 (February 2010). DOI: 10.1214/09-AOS703

Abstract

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on [0, 1]. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a Hölder class with regularity β, for all positive β, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.

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Judith Rousseau. "Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density." Ann. Statist. 38 (1) 146 - 180, February 2010. https://doi.org/10.1214/09-AOS703

Information

Published: February 2010
First available in Project Euclid: 31 December 2009

zbMATH: 1181.62047
MathSciNet: MR2589319
Digital Object Identifier: 10.1214/09-AOS703

Subjects:
Primary: 62G07 , 62G20

Keywords: adaptive estimation , Bayesian nonparametric , ‎kernel‎ , mixtures of Betas , rates of convergence

Rights: Copyright © 2010 Institute of Mathematical Statistics

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Vol.38 • No. 1 • February 2010
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