On clustering procedures and nonparametric mixture estimation

TL;DR

This paper introduces a nonparametric method for estimating conditional densities in mixture models with covariates, using clustering to improve accuracy and analyzing the convergence rates and clustering performance.

Contribution

It proposes a novel approach combining clustering and kernel density estimation for nonparametric mixture models with covariates, along with theoretical convergence analysis.

Findings

Optimal convergence rates are derived for the estimation method.

Upper bounds are established for clustering misclassification error.

Applications demonstrate effectiveness on electricity data and simulations.

Abstract

This paper deals with nonparametric estimation of conditional den-sities in mixture models in the case when additional covariates are available. The proposed approach consists of performing a prelim-inary 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 ap-plied 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 elec-tricity distribution data and simulated data are presented.