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 available. 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 estimated using kernel density estimates applied separately to each cluster. We investigate the expected $L_{1}$-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 algorithms 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.