Bias Reduction for nonparametric Estimators applied to functional Data Analysis

TL;DR

This paper tackles the elevated bias of kernel-based nonparametric estimators in functional data analysis due to one-sided kernels. It introduces a general bias-reduction framework that forms a linear combination of pilot estimators across a grid of regularisation parameters to cancel leading bias terms, preserving the linear-smoother structure. The authors establish that the bias can be reduced from order $h$ to order $h^2$ (and similarly for density components with bandwidths), while maintaining the same order of variance, and derive asymptotic normality and uniform rates for the bias-reduced estimators. The finite-sample study confirms substantial bias reduction with competitive variance, and the work discusses how to choose the regularisation grid using ideas from optimal experimental design. Overall, the approach provides a broad, practically implementable method to enable reliable inference for nonparametric functional-data estimators, including regression, conditional distribution, and conditional density tasks.

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.

Theorems & Definitions (13)

• Theorem 2.1
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4
• Remark 3.5
• Lemma 3.6
• Remark 3.7
• Theorem 3.8
• Theorem 3.9