Asymptotics for M-type smoothing splines with non-smooth objective functions
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M-type smoothing splines are a broad class of spline estimators that include the popular least-squares smoothing spline as a particular case, but also spline estimators that are less susceptible to outlying observations and model-misspecification. Available asymptotic theory, however, only covers smoothing spline estimators based on smooth objective functions and consequently leaves out frequently-used resistant estimators such as least-absolute deviations and Huber-types. We provide a general treatment in this paper and, assuming only the convexity of the objective function and some minor regularity of the error, show that the least-squares (super-)convergence rates can be extended to general M-type estimators. Auxiliary scale estimates may also be handled with slightly stronger assumptions. We further obtain optimal rates of convergence for the derivatives, which have not been obtained outside the least-squares framework.
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