Root n bandwidths selectors in multivariate kernel density estimation

• 成果类型: Article
• 署名作者: Wu, TJ; Tsai, MH
• 署名单位: National Cheng Kung University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-004-0357-8
• 发表日期: 2004
• 页码: 537-558
• 关键词: cross-validation

摘要

Based on a random sample of size n from an unknown d-dimensional density f, the problem of selecting the bandwidths in kernel estimation of f is investigated. The optimal root n relative convergence rate for bandwidth selection is established and the information bounds in this convergence are given, and a stabilized bandwidth selector (SBS) is proposed. It is known that for all d the bandwidths selected by the least squares cross-validation (LSCV) have large sample variations. The proposed SBS, as an improvement of LSCV, will reduce the variation of LSCV without significantly inflating its bias. The key idea of the SBS is to modify the d-dimensional sample characteristic function beyond some cut-off frequency in estimating the integrated squared bias. It is shown that for all d and sufficiently smooth f and kernel, if the bandwidth in each coordinate direction varies freely, then the multivariate SBS is asymptotically normal with the optimal root n relative convergence rate and achieves the (conjectured) lower bound on the covariance matrix.