Adaptive density estimation for clustering with Gaussian mixtures

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 $\beta$-Hölder with moment and tail conditions are considered. We show that this penalized estimator is minimax adaptive to the $\beta$ regularity of such densities in the Hellinger sense.