Open Access
December, 1991 Bandwidth Selection for Kernel Density Estimation
Shean-Tsong Chiu
Ann. Statist. 19(4): 1883-1905 (December, 1991). DOI: 10.1214/aos/1176348376


The problem of automatic bandwidth selection for a kernel density estimator is considered. It is well recognized that the bandwidth estimate selected by the least squares cross-validation is subject to large sample variation. This difficulty limits the application of the cross-validation estimate. Based on characteristic functions, an important expression for the cross-validation bandwidth estimate is obtained. The expression clearly points out the source of variation. To stabilize the variation, a simple bandwidth selection procedure is proposed. It is shown that the stabilized bandwidth selector gives a strongly consistent estimate of the optimal bandwidth. Under commonly used smoothness conditions, the stabilized bandwidth estimate has a faster convergence rate than the convergence rate of the cross-validation estimate. For sufficiently smooth density functions, it is shown that the stabilized bandwidth estimate is asymptotically normal with a relative convergence rate $n^{-1/2}$ instead of the rate $n^{-1/10}$ of the cross-validation estimate. A plug-in estimate and an adjusted plug-in estimate are also proposed, and their asymptotic distributions are obtained. It is noted that the plug-in estimate is asymptotically efficient. The adjusted plug-in bandwidth estimate and the stabilized bandwidth estimate are shown to be asymptotically equivalent. The simulation results verify that the proposed procedures perform much better than the cross-validation for finite samples.


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Shean-Tsong Chiu. "Bandwidth Selection for Kernel Density Estimation." Ann. Statist. 19 (4) 1883 - 1905, December, 1991.


Published: December, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0749.62022
MathSciNet: MR1135154
Digital Object Identifier: 10.1214/aos/1176348376

Primary: 62G99
Secondary: 62E20 , 62F10

Keywords: Bandwidth selection , Characteristic function , cross-validation , kernel density estimation , plug-in method

Rights: Copyright © 1991 Institute of Mathematical Statistics

Vol.19 • No. 4 • December, 1991
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