Robust estimation for functional quadratic regression models

Abstract

Functional quadratic regression models postulate a polynomial relationship rather than a linear one between a scalar response and a functional covariate. As in functional linear regression, vertical and especially high–leverage outliers may affect the classical estimators. For that reason, 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. Consistency, asymptotic robustness as well as an expression for the influence function of the related functionals are derived when the covariates have a finite–dimensional expansion. The results of a numerical study show the benefits of the robust proposal over the one based on sample principal directions and least squares for the considered contaminating scenarios. 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 method behaves very similarly to the robust one computed with all the data.