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Journal of Probability and Statistics
Volume 2015, Article ID 242683, 21 pages
Research Article

Optimal Bandwidth Selection for Kernel Density Functionals Estimation

Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA

Received 10 April 2015; Revised 19 June 2015; Accepted 21 June 2015

Academic Editor: Ricardas Zitikis

Copyright © 2015 Su Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 with appropriate choice of and furthermore equality of scale and location tests can be transformed to comparisons of the density functionals among populations. can be estimated nonparametrically via kernel density functionals estimation (KDFE). However, the optimal bandwidth selection for KDFE of 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 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.