  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.
