Robust estimation for functional quadratic regression models

TL;DR

This paper introduces a robust estimation method for functional quadratic regression models that effectively handles outliers, improving reliability over traditional least squares approaches.

Contribution

It develops a new robust estimation procedure combining principal component and regression estimators, with proven Fisher-consistency for functional quadratic models.

Findings

Robust estimators outperform classical methods in the presence of outliers.

Numerical studies demonstrate the effectiveness of the proposed method.

Application to real data shows similar results between classical and robust methods after outlier removal.

Abstract

Functional quadratic regression models postulate a polynomial relationship between a scalar response rather than a linear one. As in functional linear regression, vertical and specially high-leverage outliers may affect the classical estimators. For that reason, the proposal of robust procedures providing reliable estimators in such situations is an important issue. Taking into account that the functional polynomial model is equivalent to a regression model that is a polynomial of the same order in the functional principal component scores of the predictor processes, our proposal combines robust estimators of the principal directions with robust regression estimators based on a bounded loss function and a preliminary residual scale estimator. Fisher-consistency of the proposed method is derived under mild assumptions. The results of a numerical study show, for finite samples, the benefits of the robust proposal over the one based on sample principal directions and least squares. The usefulness of the proposed approach is also illustrated through the analysis of a real data set which reveals that when the potential outliers are removed the classical and robust methods behave very similarly.