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October 2016 From sparse to dense functional data and beyond
Xiaoke Zhang, Jane-Ling Wang
Ann. Statist. 44(5): 2281-2321 (October 2016). DOI: 10.1214/16-AOS1446


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.


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Xiaoke Zhang. Jane-Ling Wang. "From sparse to dense functional data and beyond." Ann. Statist. 44 (5) 2281 - 2321, October 2016.


Received: 1 May 2015; Revised: 1 January 2016; Published: October 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1349.62161
MathSciNet: MR3546451
Digital Object Identifier: 10.1214/16-AOS1446

Primary: 62G20
Secondary: 62G05 , 62G08

Keywords: $L^{2}$ convergence , asymptotic normality , local linear smoothing , Uniform convergence , weighing schemes

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 5 • October 2016
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