Adaptive density estimation for clustering with Gaussian mixtures

¹ Institut de Mathématiques de Toulouse, INSA de Toulouse, Université de Toulouse, INSA de Toulouse, 135 avenue de Rangueil, 31077 Toulouse Cedex 4, France
cathy.maugis@insa-toulouse.fr
² Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France
bertrand.michel@upmc.fr

Received: 17 October 2011
Revised: 18 May 2012

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.