Robust functional regression based on principal components

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 and consistency for finite-dimensional processes under mild regularity conditions. Their influence function is also studied. 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.