The Annals of Statistics

Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data

Yehua Li and Tailen Hsing

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We consider nonparametric estimation of the mean and covariance functions for functional/longitudinal data. Strong uniform convergence rates are developed for estimators that are local-linear smoothers. Our results are obtained in a unified framework in which the number of observations within each curve/cluster can be of any rate relative to the sample size. We show that the convergence rates for the procedures depend on both the number of sample curves and the number of observations on each curve. For sparse functional data, these rates are equivalent to the optimal rates in nonparametric regression. For dense functional data, root-n rates of convergence can be achieved with proper choices of bandwidths. We further derive almost sure rates of convergence for principal component analysis using the estimated covariance function. The results are illustrated with simulation studies.

Article information

Ann. Statist. Volume 38, Number 6 (2010), 3321-3351.

First available in Project Euclid: 20 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J05: Linear regression
Secondary: 62G20: Asymptotic properties 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11]

Almost sure convergence functional data analysis kernel local polynomial nonparametric inference principal components


Li, Yehua; Hsing, Tailen. Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 (2010), no. 6, 3321--3351. doi:10.1214/10-AOS813.

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