  Penalized spline smoothing is a popular and flexible method of obtaining
estimates in nonparametric regression but the classical least-squares criterion
is highly susceptible to model deviations and atypical observations. Penalized
spline estimation with a resistant loss function is a natural remedy, yet to
this day the asymptotic properties of M-type penalized spline estimators have
not been studied. We show in this paper that M-type penalized spline estimators
achieve the same rates of convergence as their least-squares counterparts, even
with auxiliary scale estimation. We further find theoretical justification for
the use of a small number of knots relative to the sample size. We illustrate
the benefits of M-type penalized splines in a Monte-Carlo study and two
real-data examples, which contain atypical observations.
