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A non asymptotic penalized criterion for Gaussian mixture model selection. (English) Zbl 1395.62162
Summary: Specific Gaussian mixtures are considered to solve simultaneously variable selection and clustering problems. A non asymptotic penalized criterion is proposed to choose the number of mixture components and the relevant variable subset. Because of the non linearity of the associated Kullback-Leibler contrast on Gaussian mixtures, a general model selection theorem for maximum likelihood estimation proposed by P. Massart [Concentration inequalities and model selection. Ecole d’Eté de Probabilités de Saint-Flour XXXIII – 2003. Berlin: Springer (2007; Zbl 1170.60006)] is used to obtain the penalty function form. This theorem requires to control the bracketing entropy of Gaussian mixture families. The ordered and non-ordered variable selection cases are both addressed in this paper.

62H30 Classification and discrimination; cluster analysis (statistical aspects)
62G07 Density estimation
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