Electronic Journal of Statistics

Adaptive density estimation based on a mixture of Gammas

Natalia Bochkina and Judith Rousseau

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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.

Article information

Electron. J. Statist., Volume 11, Number 1 (2017), 916-962.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Adaptive estimation Bayesian nonparametric estimation density estimation Dirichlet process local Hölder class mixture prior rate of contraction unbounded density

Creative Commons Attribution 4.0 International License.


Bochkina, Natalia; Rousseau, Judith. Adaptive density estimation based on a mixture of Gammas. Electron. J. Statist. 11 (2017), no. 1, 916--962. doi:10.1214/17-EJS1247. https://projecteuclid.org/euclid.ejs/1490688318

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