Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally β-Hölder with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the β regularity of such densities in the Hellinger sense.
Mots clés : rate adaptive density estimation, gaussian mixture clustering, hellinger risk, non asymptotic model selection
@article{PS_2013__17__698_0,
author = {Maugis-Rabusseau, C. and Michel, B.},
title = {Adaptive density estimation for clustering with gaussian mixtures},
journal = {ESAIM: Probability and Statistics},
pages = {698--724},
publisher = {EDP-Sciences},
volume = {17},
year = {2013},
doi = {10.1051/ps/2012018},
mrnumber = {3126158},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/ps/2012018/}
}
TY - JOUR AU - Maugis-Rabusseau, C. AU - Michel, B. TI - Adaptive density estimation for clustering with gaussian mixtures JO - ESAIM: Probability and Statistics PY - 2013 SP - 698 EP - 724 VL - 17 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2012018/ DO - 10.1051/ps/2012018 LA - en ID - PS_2013__17__698_0 ER -
%0 Journal Article %A Maugis-Rabusseau, C. %A Michel, B. %T Adaptive density estimation for clustering with gaussian mixtures %J ESAIM: Probability and Statistics %D 2013 %P 698-724 %V 17 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2012018/ %R 10.1051/ps/2012018 %G en %F PS_2013__17__698_0
Maugis-Rabusseau, C.; Michel, B. Adaptive density estimation for clustering with gaussian mixtures. ESAIM: Probability and Statistics, Tome 17 (2013), pp. 698-724. doi : 10.1051/ps/2012018. http://archive.numdam.org/articles/10.1051/ps/2012018/
[1] , and , Slope heuristics: overview and implementation. Stat. Comput. 22 (2011) 455-470. | MR
[2] , A new lower bound for multiple hypothesis testing. IEEE Trans. Inform. Theory. 51 (2005) 1611-1615. | MR | Zbl
[3] and , A course in approximation theory, Graduate Studies in Mathematics, vol. 101 of Amer. Math. Soc. Providence, RI (2009). | MR | Zbl
[4] , and , Posterior consistency of Dirichlet mixtures in density estimation. Ann. Stat. 27 (1999) 143-158. | MR | Zbl
[5] and , Entropy and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Stat. 29 (2001) 1233-1263,. | MR | Zbl
[6] and , Posterior convergence rates of Dirichlet mixtures at smooth densities. Ann. Stat. 35 (2007) 697-723. | MR | Zbl
[7] , Abstract inference. John Wiley and Sons Inc., New York (1981). | MR | Zbl
[8] and , Nonlinear approximation using Gaussian kernels. J. Functional Anal. 259 (2010) 203-219. | MR | Zbl
[9] , Clustering algorithms, Probab. Math. Stat. John Wiley and Sons, New York-London-Sydney (1975). | MR | Zbl
[10] , and , The elements of statistical learning, Data mining, inference, and prediction. Statistics. Springer, New York, 2nd edition (2009). | MR | Zbl
[11] W. Kruijer, J. Rousseau and A van der Vaart, Adaptive Bayesian Density Estimation with Location-Scale Mixtures. Electron. J. Statist. 4 (2010) 1225-1257. | MR
[12] , Mixtures Models: Theory, Geometry and Applications. IMS, Hayward, CA (1995). | Zbl
[13] , Concentration Inequalities and Model Selection. École d'été de Probabilités de Saint-Flour, 2003. Lect. Notes Math. Springer (2007). | MR | Zbl
[14] and , Adaptive density estimation for clustering with Gaussian mixtures (2011). arXiv:1103.4253v2.
[15] and , Data-driven penalty calibration: a case study for Gaussian mixture model selection. ESAIM: PS 15 (2011) 320-339. | Numdam | MR
[16] and , A non asymptotic penalized criterion for Gaussian mixture model selection. ESAIM: PS 15 (2011) 41-68. | Numdam | MR
[17] and , Finite Mixture Models. Wiley (2000). | MR | Zbl
[18] , Introduction to nonparametric estimation. Statistics. Springer, New York (2009). | MR | Zbl
[19] , Minimax estimation of the mean of a normal distribution with known variance. Ann. Math. Stat. 21 (1950) 218-230. | MR | Zbl
Cité par Sources :
