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Adaptive density estimation for clustering with Gaussian mixtures

Published online by Cambridge University Press:  04 November 2013

C. Maugis-Rabusseau  and
B. Michel
C. Maugis-Rabusseau
Affiliation:
Institut de Mathématiques de Toulouse, INSA de Toulouse, Université de Toulouse, INSA de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France. [email protected]
B. Michel
Affiliation:
Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France; [email protected]
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Abstract

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.

Keywords

Rate adaptive density estimationgaussian mixture clusteringhellinger risknon asymptotic model selection
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 17 , 2013 , pp. 698 - 724
Copyright
© EDP Sciences, SMAI, 2013

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