Nonparametric Estimation of Variance Function for Functional Data

TL;DR

This paper develops a nonparametric kernel estimator for the variance function in functional data analysis, showing its asymptotic properties and superior performance over residual-based methods through simulations.

Contribution

It introduces an asymptotic analysis of a kernel estimator for variance functions in functional data with unknown means, highlighting the impact of mean smoothness.

Findings

Estimator based on residuals outperforms that based on conditional second moments.

Asymptotic results reveal the influence of mean function smoothness on convergence rates.

Simulation studies confirm the effectiveness of the proposed residual-based estimator.

Abstract

This article investigates nonparametric estimation of variance functions for functional data when the mean function is unknown. We obtain asymptotic results for the kernel estimator based on squared residuals. Similar to the finite dimensional case, our asymptotic result shows the smoothness of the unknown mean function has an effect on the rate of convergence. Our simulaton studies demonstrate that estimator based on residuals performs much better than that based on conditional second moment of the responses.