FROM SPARSE TO DENSE FUNCTIONAL DATA AND BEYOND

• 成果类型: Article
• 署名作者: Zhang, Xiaoke; Wang, Jane-Ling
• 署名单位: University of Delaware; University of California System; University of California Davis
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1446
• 发表日期: 2016
• 页码: 2281-2321
• 关键词: varying-coefficient models; principal-components-analysis; local linear-regression; mixed-effects models; Nonparametric Regression; convergence-rates; Consistency; estimators

摘要

Nonparametric estimation of mean and covariance functions is important in functional data analysis. We investigate the performance of local linear smoothers for both mean and covariance functions with a general weighing scheme, which includes two commonly used schemes, equal weight per observation (OBS), and equal weight per subject (SUBJ), as two special cases. We provide a comprehensive analysis of their asymptotic properties on a unified platform for all types of sampling plan, be it dense, sparse or neither. Three types of asymptotic properties are investigated in this paper: asymptotic normality, L-2 convergence and uniform convergence. The asymptotic theories are unified on two aspects: (1) the weighing scheme is very general; (2) the magnitude of the number N-i of measurements for the ith subject relative to the sample size n can vary freely. Based on the relative order of Ni to n, functional data are partitioned into three types: non-dense, dense and ultra dense functional data for the OBS and SUBJ schemes. These two weighing schemes are compared both theoretically and numerically. We also propose a new class of weighing schemes in terms of a mixture of the OBS and SUBJ weights, of which theoretical and numerical performances are examined and compared.