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

TL;DR

This paper establishes uniform convergence rates for nonparametric estimators of mean and covariance functions in functional data analysis, applicable to both sparse and dense data, and extends results to principal component analysis.

Contribution

It provides a unified framework for convergence rates in nonparametric functional data estimation, accommodating varying observation densities within curves.

Findings

Optimal rates for sparse data match nonparametric regression

Root-n convergence rates achievable for dense data

Almost sure convergence results for PCA based on estimated covariance

Abstract

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.