Transferring supremum-norm rates and weak convergence of covariance kernel estimators to functional principal components

2026-07-02

AI Summary

This study investigates how to effectively transfer the supremum-norm convergence rates and weak convergence properties of covariance kernel estimators to functional principal components (FPCs). By integrating $L_2$ perturbation theory with the Nyström method and minimax lower bound analysis, the work establishes, for the first time, optimal supremum-norm convergence rates and asymptotic normality for FPC estimators. The analysis uncovers a novel phenomenon in sparse observation regimes, where eigenvalue estimation is dominated by discretization effects. These theoretical results are further extended to cross-covariance, long-run covariance, and derivative process covariance kernels. Empirical validation on temperature curves demonstrates the method’s effectiveness across the spectrum from sparse to dense observation designs.

Abstract

We show that $L_2$-perturbation theory can be used to transfer rates of convergence in the supremum norm as well as weak convergence in the space of continuous functions from covariance kernel estimators to the associated functional principle components (FPCs). As an application we obtain optimal rates of convergence in sup-norm, including minimax-lower bounds, as well as asymptotic normality for estimating the FPCs in a discrete observational model with errors under fixed, synchronous design. The sparse to dense transition which has previously been observed for mean function and covariance kernel estimators also applies to the FPCs. Surprisingly, eigenvalue estimation exhibits a discretization-dominated regime under sparse designs, too. Our results further apply to estimators of cross-covariance and long-run covariance kernels, as well as to covariance kernels of derivative processes. We also present results of numerical experiments in which we use the Nyström method to compute FPCs and eigenvalues, and give an empirical illustration to series of daily temperature curves.

Problem

• functional principal components
• covariance kernel estimation
• supremum-norm convergence
• weak convergence
• discrete observational model

Innovation

• functional principal components
• supremum-norm convergence
• weak convergence
• covariance kernel estimation
• Nyström method