ASYMPTOTICS FOR LEAST-SQUARES CROSS-VALIDATION BANDWIDTHS IN NONSMOOTH CASES

• 成果类型: Note
• 署名作者: VANES, B
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/aos/1176348790
• 发表日期: 1992
• 页码: 1647-1657
• 关键词: central limit-theorem; density estimators; selection; error

摘要

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(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 H(n) - H(n)* and H(n) - 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 of 0 of the relative distances (H(n) - H(n)*)/H(n)* and (H(n) - 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.