Electronic Journal of Statistics

Adaptive density estimation based on a mixture of Gammas

Natalia Bochkina and Judith Rousseau

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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 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1490688318

Digital Object Identifier
doi:10.1214/17-EJS1247

Mathematical Reviews number (MathSciNet)
MR3629019

Zentralblatt MATH identifier
1362.62076

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

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

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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References

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    Mathematical Reviews (MathSciNet): MR2399197
    Digital Object Identifier: doi:10.1214/07-EJS130
    Project Euclid: euclid.ejs/1210254767

The Institute of Mathematical Statistics and the Bernoulli Society

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