Robust functional regression based on principal components

TL;DR

This paper introduces a robust two-step functional linear model estimation method combining robust principal components and regression, improving resistance to outliers while maintaining efficiency and smoothness.

Contribution

It proposes a novel robust estimation procedure for functional linear models that is less sensitive to atypical data points and includes a transformation for better estimator behavior.

Findings

Estimators are Fisher-consistent at elliptical distributions.

Proposed methods are robust against outliers in simulations.

Estimators produce smooth, efficient, and reliable functional regression results.

Abstract

Functional data analysis is a fast evolving branch of modern statistics and the functional linear model has become popular in recent years. However, most estimation methods for this model rely on generalized least squares procedures and therefore are sensitive to atypical observations. To remedy this, we propose a two-step estimation procedure that combines robust functional principal components and robust linear regression. Moreover, we propose a transformation that reduces the curvature of the estimators and can be advantageous in many settings. For these estimators we prove Fisher-consistency at elliptical distributions and consistency under mild regularity conditions. The influence function of the estimators is investigated as well. Simulation experiments show that the proposed estimators have reasonable efficiency, protect against outlying observations, produce smooth estimates and perform well in comparison to existing approaches.