Robust penalized spline estimation with difference penalties

TL;DR

This paper introduces a class of robust penalized spline estimators using general loss functions like Huber or Tukey, which maintain optimal asymptotic properties and perform well in finite samples, improving robustness over classical methods.

Contribution

It develops a broad class of robust P-spline estimators based on general loss functions, extending classical least-squares P-splines with strong theoretical and practical advantages.

Findings

Robust P-spline estimators retain optimal asymptotic properties.

Estimators can be computed efficiently via iterative algorithms.

Numerical studies show excellent finite-sample performance.

Abstract

Penalized spline estimation with discrete difference penalties (P-splines) is a popular estimation method for semiparametric models, but the classical least-squares estimator is highly sensitive to deviations from its ideal model assumptions. To remedy this deficiency, a broad class of P-spline estimators based on general loss functions is introduced and studied. 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. It is shown 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.