Bootstrapping nonparametric density estimators with empirically chosen bandwidths

• 成果类型: Article
• 署名作者: Hall, P; Kang, KH
• 署名单位: Australian National University; Hankuk University Foreign Studies
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 2001
• 页码: 1443-1468
• 关键词: ACCURACY; CHOICE

摘要

We examine the way in which empirical bandwidth choice affects distributional properties of nonparametric density estimators. Two bandwidth selection methods are considered in detail: local and global plug-in rules. Particular attention is focussed on whether the accuracy of distributional bootstrap approximations is appreciably influenced by using the resample version (h) over cap*, rather than the sample version (h) over cap, of an empirical bandwidth. It is shown theoretically that, in marked contrast to similar problems in more familiar settings, no general first-order theoretical improvement can be expected when using the resampling version. In the case of local plug-in rules, the inability of the bootstrap to accurately reflect biases of the components used to construct the bandwidth selector means that the bootstrap distribution of (h) over cap* is unable to capture some of the main properties of the distribution of (h) over cap. If the second derivative component is slightly undersmoothed then some improvements are possible through using (h) over cap*, but they would be difficult to achieve in practice. On the other hand, for global plug-in methods, both (h) over cap and (h) over cap* are such good approximations to an optimal, deterministic bandwidth that the variations of either can be largely ignored, at least at a first-order level. Thus, for quite different reasons in the two cases, the computational burden of varying an empirical bandwidth across resamples is difficult to justify.