The Annals of Statistics

Asymptotics for Least Squares Cross-Validation Bandwidths in Nonsmooth Cases

Bert van Es

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We consider the problem of bandwidth selection for kernel density estimators. Let $H_n$ denote the bandwidth computed by the least squares cross-validation method. Furthermore, let $H^\ast_n$ and $h^\ast_n$ denote the minimizers of the integrated squared error and the mean integrated squared error, respectively. The main theorem establishes asymptotic normality of $H_n - H^\ast_n$ and $H_n - h^\ast_n$, for three classes of densities with comparable smoothness properties. Apart from densities satisfying the standard smoothness conditions, we also consider densities with a finite number of jumps or kinks. We confirm the $n^{-1/10}$ rate of convergence to 0 of the relative distances $(H_n - H^\ast_n)/H^\ast_n$ and $(H_n - h^\ast_n)/h^\ast_n$ derived by Hall and Marron in the smooth case. Unexpectedly, in turns out that these relative rates of convergence are faster in the nonsmooth cases.

Article information

Ann. Statist., Volume 20, Number 3 (1992), 1647-1657.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory

Density estimation kernel estimators bandwidth selection cross-validation rates of convergence


van Es, Bert. Asymptotics for Least Squares Cross-Validation Bandwidths in Nonsmooth Cases. Ann. Statist. 20 (1992), no. 3, 1647--1657. doi:10.1214/aos/1176348790.

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