The Annals of Statistics

Bandwidth Selection for Kernel Density Estimation

Shean-Tsong Chiu

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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.

Article information

Ann. Statist., Volume 19, Number 4 (1991), 1883-1905.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G99: None of the above, but in this section
Secondary: 62F10: Point estimation 62E20: Asymptotic distribution theory

Kernel density estimation bandwidth selection cross-validation characteristic function plug-in method


Chiu, Shean-Tsong. Bandwidth Selection for Kernel Density Estimation. Ann. Statist. 19 (1991), no. 4, 1883--1905. doi:10.1214/aos/1176348376.

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