Robust estimation for semi-functional linear regression models

TL;DR

This paper develops robust estimators for semi-functional linear regression models that effectively handle high-leverage outliers, outperforming classical methods in finite samples and providing better predictions in real data applications.

Contribution

It introduces a novel robust estimation approach combining B-splines with bounded loss functions for semi-functional linear regression models, ensuring consistency and improved outlier resistance.

Findings

Robust estimators outperform least squares and Huber estimators in simulations.

Proposed methods yield better predictions for non-outlying data points.

Robust and classical estimators behave similarly when outliers are removed.

Abstract

Semi-functional linear regression models postulate a linear relationship between a scalar response and a functional covariate, and also include a non-parametric component involving a univariate explanatory variable. It is of practical importance to obtain estimators for these models that are robust against high-leverage outliers, which are generally difficult to identify and may cause serious damage to least squares and Huber-type $M$-estimators. For that reason, robust estimators for semi-functional linear regression models are constructed combining $B$-splines to approximate both the functional regression parameter and the nonparametric component with robust regression estimators based on a bounded loss function and a preliminary residual scale estimator. Consistency and rates of convergence for the proposed estimators are derived under mild regularity conditions. The reported numerical experiments show the advantage of the proposed methodology over the classical least squares and Huber-type $M$-estimators for finite samples. The analysis of real examples illustrate that the robust estimators provide better predictions for non-outlying points than the classical ones, and that when potential outliers are removed from the training and test sets both methods behave very similarly.