Mean Hellinger Distance as an Error Criterion in Univariate and Multivariate Kernel Density Estimation

Abstract

Ever since the pioneering work of Parzen the mean square error( MSE) and its integrated form (MISE) have been used as the error criteria in choosing the bandwidth matrix for multivariate kernel density estimation. More recently other criteria have been advocated as competitors to the MISE, such as the mean absolute error. In this study we define a weighted version of the Hellinger distance for multivariate densities and show that it has an asymptotic form, which is one-fourth the asymptotic MISE under weak smoothness conditions on the multivariate density f. In addition the proposed criteria give rise to a new data-dependent bandwidth matrix selector. The performance of the new data-dependent bandwidth matrix selector is compared with other well known bandwidth matrix selectors such as the least squared cross validation (LSCV) and the plug-in (HPI) through simulation. We derived a closed form formula for the mean Hellinger distance (MHD) in the univariate case. We also compared via simulation mean weighted Hellinger distance (MWHD) and the asymptotic MWHD, and the MISE and the asymptotic MISE for both univariate and bivariate cases for various densities and sample sizes