  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.
