Electronic Journal of Statistics

Adaptive density estimation based on a mixture of Gammas

Natalia Bochkina and Judith Rousseau

Full-text: Open access


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

Permanent link to this document

Digital Object Identifier

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

Export citation


  • [1] Canale, A. and Blasi, P. D. (2017). Posterior asymptotics of nonparametric location-scale mixtures for multivariate density estimation., Bernoulli, 23(1):379–404.
  • [2] Copsey, K. and Webb, A. (2003). Bayesian gamma mixture model approach to radar target recognition., IEEE Transactions on Aerospace and Electronic Systems, 39(4):1201–1217.
  • [3] Ghosal, S., Ghosh, J. K., and van der Vaart, A. (2000). Convergence rates of posterior distributions., Ann. Statist., 28:500–531.
  • [4] Ghosal, S. and van der Vaart, A. (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities., Ann. Statist., 35(2):697–723.
  • [5] Ghosal, S. and van der Vaart, A. W. (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities., Ann. Statist., 29(5):1233–1263.
  • [6] Kalli, M., Griffin, J. E., and Walker, S. G. (2011). Slice sampling mixture models., Statistics and Computing, 21:93–105.
  • [7] Kruijer, W., Rousseau, J., and van der Vaart, A. (2010). Adaptive Bayesian density estimation with location-scale mixtures., Electron. J. Stat., 4:1225–1257.
  • [8] Li, X. and Chen, C.-P. (2007). Inequalities for the Gamma function., Journal of inequalities in pure and applied mathematics, 8(1):Article 28.
  • [9] Maugis, C. and Michel, B. (2013). Adaptive density estimation using finite Gaussian mixtures., ESAIM: Probability and Statistics, 17:698–724.
  • [10] McVinish, R., Rousseau, J., and Mengersen, K. (2009). Bayesian goodness-of-fit testing with mixtures of triangular distributions., Scandinavian Journ. Statist., 36:337–354.
  • [11] Rousseau, J. (2010). Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparamatric estimation of the density., Ann. Statist., 38(1):146–180.
  • [12] Scricciolo, C. (2009). Adaptive Bayesian density estimation in Lp-metrics with Pitman-Yor or normalized inverse-Gaussian process kernel mixtures., Bayesian Analysis, 9:475–520.
  • [13] Scricciolo, C. (2011). Posterior rates of convergence for Dirichlet mixtures of exponential power densities., Electronic Journal of Statistics, 5:270–308.
  • [14] Shen, W., Tokdar, S., and Ghosal, S. (2013). Adaptive Bayesian multivariate density estimation with Dirichlet mixtures., Biometrika, pages 1–18.
  • [15] Tapattu, S., Tellambura, C., and Jiang, H. (2011). A mixture gamma distribution to model the SNR of wireless channels., IEEE Wireless Communications, 12(10):4193–4203.
  • [16] Wiper, M., Insua, D. R., and Ruggeri, F. (2001). Mixtures of Gamma distributions with applications., Journal of Computational and Graphical Statistics, 10(3):440–454.
  • [17] Wu, Y. and Ghosal, S. (2008). Kullback Leibler property of kernel mixture priors in Bayesian density estimation., Electronic Journal of Statistics, 2:298–331.