  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.
