On the minimisation of L(p) error in mode estimation

• 成果类型: Article
• 署名作者: Grund, B; Hall, P
• 署名单位: Australian National University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1995
• 页码: 2264-2284
• 关键词: smoothing parameter; DENSITY-ESTIMATION; bootstrap choice; kernel

摘要

We show that, for L(p) convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density, finiteness of the pth moment of the underlying distribution is both necessary and sufficient. The basic requirement of existence of finite variance has been overlooked by statisticians, who have earlier considered mean square convergence of nonparametric mode estimators; they have focussed on mean squared error of the asymptotic distribution, rather than an asymptotic mean squared error. The effect of bandwidth choice on the rate of L(p) convergence is analysed, and smoothed bootstrap methods are used to develop an empirical approximation to the L(p) measure of error. The resulting bootstrap estimator of L(p) error may be minimised with respect to the bandwidth of the nonparametric density estimator, and in this way an empirical rule may he developed for selecting the bandwidth for mode estimation. Particular attention is devoted to the problem of selecting the appropriate amount of smoothing in the bootstrap algorithm.