Asymptotics for Least Squares Cross-Validation Bandwidths in Nonsmooth Cases

We consider the problem of bandwidth selection for kernel density estimators. Let Hₙ denote the bandwidth computed by the least squares cross-validation method. Furthermore, let H^ₙ and h^ₙ denote the minimizers of the integrated squared error and the mean integrated squared error, respectively. The main theorem establishes asymptotic normality of Hₙ - H^ₙ and Hₙ - h^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 (Hₙ - H^ₙ)/H^ₙ and (Hₙ - h^ₙ)/h^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.