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

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
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