Open Access
2017 Adaptive density estimation based on a mixture of Gammas
Natalia Bochkina, Judith Rousseau
Electron. J. Statist. 11(1): 916-962 (2017). DOI: 10.1214/17-EJS1247


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 Hölder 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.

One of the novel feature of the paper is that these results hold for densities with polynomial tails. Similar results are obtained using a hierarchical Bayesian model based on the mixture of inverse gamma densities which can be used to estimate adaptively densities with very heavy tails, including Cauchy density.


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Natalia Bochkina. Judith Rousseau. "Adaptive density estimation based on a mixture of Gammas." Electron. J. Statist. 11 (1) 916 - 962, 2017.


Received: 1 May 2016; Published: 2017
First available in Project Euclid: 28 March 2017

zbMATH: 1362.62076
MathSciNet: MR3629019
Digital Object Identifier: 10.1214/17-EJS1247

Primary: 62G07
Secondary: 62G20

Keywords: adaptive estimation , Bayesian nonparametric estimation , Density estimation , Dirichlet process , local Hölder class , mixture prior , rate of contraction , unbounded density

Vol.11 • No. 1 • 2017
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