Bias Reduction for nonparametric Estimators applied to functional Data Analysis

TL;DR

This paper introduces bias-corrected kernel estimators for functional data analysis that reduce bias without increasing variance, enabling more accurate statistical inference.

Contribution

It proposes a novel bias correction method for kernel estimators in functional data models, improving their asymptotic properties and finite sample performance.

Findings

Bias of the new estimators is of smaller order than traditional methods.

Asymptotic normality of the bias-corrected estimators is established.

Simulation results demonstrate improved finite sample performance.

Abstract

Compared to nonparametric estimators in the multivariate setting, kernel estimators for functional data models have a larger order of bias. This is problematic for constructing confidence regions or statistical tests since the bias might not be negligible. It stems from the fact that one sided kernels are used where already the first moment of the kernel is different from 0. It cannot be cured by assuming the existence of higher order derivatives. In the following, we propose bias corrected estimators based on the idea in \cite{Cheng2018} which still have an appealing structure, but have a bias of smaller order as in multiple regression settings while the variance is of the same order of magnitude as before. In addition we show asymptotic normality of such estimators and derive uniform rates. The performance of the estimator in finite samples is in addition checked in a simulation study.