Asymptotics for M-type smoothing splines with non-smooth objective functions

M-type smoothing splines are a broad class of spline estimators that include the popular least-squares but also less susceptible to outlying observations and model misspecification. However, available asymptotic theory only covers based on smooth objective functions consequently leaves out frequently used resistant such as quantile Huber-type splines. We provide general treatment in this paper and, assuming convexity function, show (super-)convergence rates can be extended whose properties have not been hitherto described. further auxiliary scale estimates may handled under significantly weaker assumptions than those found literature we establish optimal convergence for derivatives, which obtained outside framework. A simulation study real-data example illustrate competitive performance non-smooth relation regular data their superior contain anomalies.