Volume 17, 2013
|Page(s)||698 - 724|
|Published online||04 November 2013|
Adaptive density estimation for clustering with Gaussian mixtures
Institut de Mathématiques de Toulouse, INSA de Toulouse,
Université de Toulouse, INSA de
Toulouse, 135 avenue de Rangueil, 31077
Toulouse Cedex 4,
2 Laboratoire de Statistique Théorique et Appliquée, Université Pierre et Marie Curie - Paris 6, 4 place Jussieu, 75252 Paris Cedex 05, France
Received: 17 October 2011
Revised: 18 May 2012
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.
Mathematics Subject Classification: 62G07 / 62G20
Key words: Rate adaptive density estimation / gaussian mixture clustering / hellinger risk / non asymptotic model selection
© EDP Sciences, SMAI, 2013
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