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Adaptive density estimation based on a mixture of Gammas

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Original languageEnglish
Pages (from-to)916-962
Number of pages47
JournalElectronic Journal of Statistics
Volume11
Issue number1
Early online date28 Mar 2017
DOIs
Publication statusPublished - Apr 2017

Abstract

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local H\"older densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness.

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