M-type penalized splines for functional linear regression
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Functional data analysis is a fast evolving branch of modern statistics, yet despite the popularity of the functional linear model in recent years, current estimation procedures either suffer from lack of robustness or are computationally burdensome. To address these shortcomings, we propose a flexible family of lower-rank smoothers that combines penalized splines and M-estimation. Under a condition on the design matrix, these estimators exhibit the same asymptotic properties as the corresponding least-squares estimators, while being considerably more reliable in the presence of outliers. The proposed methods easily generalize to functional models that include scalar covariates or nonparametric components, thus providing a wide framework of estimation. The finite-sample performance of the proposed family of estimators is illustrated on an extensive simulation study as well as a real data set, where it is found that the proposed estimators can combine high efficiency with protection against outliers, and produce smooth estimates that compare favourably with existing approaches, robust and non-robust alike.
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