## The Annals of Statistics

### Asymptotics for Least Squares Cross-Validation Bandwidths in Nonsmooth Cases

Bert van Es

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348790

Digital Object Identifier
doi:10.1214/aos/1176348790

Mathematical Reviews number (MathSciNet)
MR1186271

Zentralblatt MATH identifier
0763.62025

JSTOR
links.jstor.org

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176348790