Bias Reduction for nonparametric Estimators applied to functional Data Analysis

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.