Root n bandwidth selectors for kernel estimation of density derivatives

• 成果类型: Article
• 署名作者: Wu, TJ
• 署名单位: National Dong Hwa University
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.2307/2965702
• 发表日期: 1997
• 页码: 536-547
• 关键词: squares cross-validation; error properties; CHOICE

摘要

Based on a random sample of size n from an unknown density f on the real line, the problem of adaptively selecting the bandwidth in kernel estimation of f((k)) is investigated, where f((k)) denotes the kth derivative of f. The estimates of density derivatives can be used to evaluate modes and infection points of f and can be applied to the estimation of scores in certain additive models and to the empirical verification of the law of demand in econometrics. For all k, the information bounds for bandwidth selectors are given, and two types of adaptive bandwidth selectors are proposed. A careful analysis of the kernel estimate of the integrated squared bias (ISB) term, contained in a generalization of smoothed cross-validation, indicates that the ''sync kernel'' is the proper choice for estimating ISB and, consequently, leads to the first type of selector. The second type of selector is based on the plug-in method, which involves expanding ISB up to suitable order and estimating integrated squared density derivatives. It is shown that for all k and sufficiently smooth f and kernel, both of the proposed selectors are asymptotically normal with the optimal O-p(n(-1/2)) relative convergence rate and achieve the (conjectured) information bound. This asymptotically optimal performance requires stronger smoothness assumptions on f and kernel for larger k. For k = 1 and 2, extensive simulation studies have been done, and the excellent performances of the proposed selectors at practical sample sizes are clearly demonstrated. In particular, the proposed selectors perform conclusively better than the one selected by cross-validation.