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

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

2010-01-11

arxiv.org/abs/1001.1615v1

Annals of Statistics 2010, Vol. 38, No. 1, 146-180

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS703 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS703

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 H\"{o}lder class with regularity $\beta$, for all positive $\beta$, 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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