From sparse to dense functional data and beyond. (English) Zbl 1349.62161

Zhang, Xiaoke; Wang, Jane-Ling

Ann. Stat. 44, No. 5, 2281-2321 (2016).

Summary: 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 \(i\)th subject relative to the sample size \(n\) can vary freely. Based on the relative order of \(N_{i}\) 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.

MSC:

62G20	Asymptotic properties of nonparametric inference
62G08	Nonparametric regression and quantile regression
62G05	Nonparametric estimation

Keywords:

local linear smoothing; asymptotic normality; \(L^{2}\) convergence; uniform convergence; weighing schemes