Robust penalized estimators for functional linear regression

Functional data analysis is a fast evolving branch of statistics. Estimation procedures for the popular functional linear model either suffer from lack of robustness or are computationally burdensome. To address these shortcomings, a flexible family of penalized lower-rank estimators based on a bounded loss function is proposed. The proposed class of estimators is shown to be consistent and can attain high rates of convergence with respect to prediction error under weak regularity conditions. These results can be generalized to higher dimensions under similar assumptions. The finite-sample performance of the proposed family of estimators is investigated by a Monte-Carlo study which shows that these estimators reach high efficiency while offering protection against outliers. The proposed estimators compare favorably to existing approaches robust as well as non-robust alternatives. The good performance of the method is also illustrated on a complex real dataset.