Asymptotics for Least Squares Cross-Validation Bandwidths in Nonsmooth Cases

Abstract

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.