The Annals of Statistics

Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

Judith Rousseau

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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.

Article information

Source
Ann. Statist., Volume 38, Number 1 (2010), 146-180.

Dates
First available in Project Euclid: 31 December 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1262271612

Digital Object Identifier
doi:10.1214/09-AOS703

Mathematical Reviews number (MathSciNet)
MR2589319

Zentralblatt MATH identifier
1181.62047

Subjects
Primary: 62G07: Density estimation 62G20: Asymptotic properties

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

Citation

Rousseau, Judith. Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density. Ann. Statist. 38 (2010), no. 1, 146--180. doi:10.1214/09-AOS703. https://projecteuclid.org/euclid.aos/1262271612


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