Nonparametric Estimation of Variance Function for Functional Data

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.

💡 Research Summary

This paper addresses the problem of non‑parametric estimation of a variance function when the data consist of functional covariates and the mean function is unknown. In many functional data analyses the focus lies on estimating the mean curve, while the variability of the response conditional on the functional predictor is often assumed to be homoscedastic or is estimated by directly smoothing the conditional second moment (E