Robust penalized spline estimation with difference penalties
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Penalized spline estimation with discrete difference 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 deficiency 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 efficiently through a simple adaptation of well-established iterative least squares algorithms and exhibit excellent performance even in finite samples, as evidenced by a numerical study and a real-data example.
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