期刊
ANNALS OF STATISTICS
卷 44, 期 5, 页码 2281-2321出版社
INST MATHEMATICAL STATISTICS-IMS
DOI: 10.1214/16-AOS1446
关键词
Local linear smoothing; asymptotic normality; L-2 convergence; uniform convergence; weighing schemes
资金
- NSF [DMS-09-06813]
- Direct For Mathematical & Physical Scien [1228369] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [1228369] Funding Source: National Science Foundation
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [1512975] Funding Source: National Science Foundation
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.
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