M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline but also spline estimators that are less susceptible to outlying observations and model-misspecification. However, available asymptotic theory only covers smoothing spline estimators based on smooth objective functions and consequently leaves out frequently used resistant estimators such as quantile and Huber-type smoothing splines. We provide a general treatment in this paper and, assuming only the convexity of the objective function, show that the least-squares (super-)convergence rates can be extended to M-type estimators whose asymptotic properties have not been hitherto described. We further show that auxiliary scale estimates may be handled under significantly weaker assumptions than those found in the literature and we establish optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework. A simulation study and a real-data example illustrate the competitive performance of non-smooth M-type splines in relation to the least-squares spline on regular data and their superior performance on data that contain anomalies.

Functional data analysis is a fast evolving branch of modern statistics and the functional linear model has become popular in recent years. However, most estimation methods for this model rely on generalized least squares procedures and therefore are sensitive to atypical observations. To remedy this, we propose a two-step estimation procedure that combines robust functional principal components and robust linear regression. Moreover, we propose a transformation that reduces the curvature of the estimators and can be advantageous in many settings. For these estimators we prove Fisher-consistency at elliptical distributions and consistency under mild regularity conditions. The influence function of the estimators is investigated as well. Simulation experiments show that the proposed estimators have reasonable efficiency, protect against outlying observations, produce smooth estimates and perform well in comparison to existing approaches.

Penalized spline estimation with discrete dierence penalties (P-splines) is a popular estimation method in semiparametric models, but the classical least-squares estimator is susceptible to gross errors and other model deviations. To remedy this deciency we introduce and study a broad class of P-spline estimators based on general loss functions. Robust estimators are obtained by well-chosen loss functions, such as the Huber or Tukey loss function. A preliminary scale estimator can also be included in the loss function. We show in this paper that this class of P-spline estimators enjoys the same optimal asymptotic properties as least-squares P-splines, thereby providing strong theoretical motivation for its use. The proposed estimators may be computed very eciently through a simple adaptation of well-established iterative least squares algorithms and exhibit excellent performance even in nite samples, as evidenced by a numerical study and a real-data example.

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Functional data analysis is a fast evolving branch of modern statistics. 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 suitable conditions on the design, 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 can easily be generalized to functional models with additional functional predictors, scalar covariates or nonparametric components, thus providing a wide framework of estimation. Empirical investigation shows 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.