UNIFORM CONVERGENCE RATES FOR NONPARAMETRIC REGRESSION AND PRINCIPAL COMPONENT ANALYSIS IN FUNCTIONAL/LONGITUDINAL DATA

• 成果类型: Article
• 署名作者: Li, Yehua; Hsing, Tailen
• 署名单位: University System of Georgia; University of Georgia; University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/10-AOS813
• 发表日期: 2010
• 页码: 3321-3351
• 关键词: sparse functional data; longitudinal data; models

摘要

We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.