β-divergence loss for the kernel density estimation with bias reduced

Abstract

In this paper, we investigate the problem of estimating the probability density function. The kernel density estimation with bias reduced is nowadays a standard technique in explorative data analysis, there is still a big dispute on how to assess the quality of the estimate and which choice of bandwidth is optimal. This framework examines the most important bandwidth selection methods for kernel density estimation in the context of with bias reduction. Normal reference, least squares cross-validation, biased cross-validation and β-divergence loss methods are described and expressions are presented. In order to assess the performance of our various bandwidth selectors, numerical simulations and environmental data are carried out.