On clustering procedures and nonparametric mixture estimation

Abstract

This paper deals with nonparametric estimation of conditional densities in mixture models in the case when additional covariates are avail- able. The proposed approach consists of performing a preliminary clustering algorithm on the additional covariates to guess the mixture component of each observation. Conditional densities of the mixture model are then es- timated using kernel density estimates applied separately to each cluster. We investigate the expected L1-error of the resulting estimates and derive optimal rates of convergence over classical nonparametric density classes provided the clustering method is accurate. Performances of clustering al- gorithms are measured by the maximal misclassification error. We obtain upper bounds of this quantity for a single linkage hierarchical clustering algorithm. Lastly, applications of the proposed method to mixture models involving electricity distribution data and simulated data are presented.