Optimal Bandwidth Selection for Kernel Density Functionals Estimation

Abstract

The choice of bandwidth is crucial to the kernel density estimation (KDE) and kernel based regression. Various bandwidth selection methods for KDE and local least square regression have been developed in the past decade. It has been known that scale and location parameters are proportional to density functionals ∫ γ (x) f 2 (x) d x with appropriate choice of γ (x) and furthermore equality of scale and location tests can be transformed to comparisons of the density functionals among populations. ∫ γ (x) f 2 (x) d x can be estimated nonparametrically via kernel density functionals estimation (KDFE). However, the optimal bandwidth selection for KDFE of ∫ γ (x) f 2 (x) d x has not been examined. We propose a method to select the optimal bandwidth for the KDFE. The idea underlying this method is to search for the optimal bandwidth by minimizing the mean square error (MSE) of the KDFE. Two main practical bandwidth selection techniques for the KDFE of ∫ γ (x) f 2 (x) d x are provided: Normal scale bandwidth selection (namely, "Rule of Thumb") and direct plug-in bandwidth selection. Simulation studies display that our proposed bandwidth selection methods are superior to existing density estimation bandwidth selection methods in estimating density functionals.