Global Convergence Rates of B - SplineM - Estimators in Nonparametric

To compensate for lack of robustness in using regression splines via the least squares principle, a robust data smoothing procedure is proposed for obtaining a robust regression spline estimator of an unknown regression function, g 0 , of a one-dimensional measurement variable. This robust regression spline estimator is computed by using the usual M-type iteration procedures proposed for linear models. A simulation study is carried out and numerical examples are given to illustrate the utility of the proposed method. Assume that g 0 is smoothed up to order r > 1=2 and denote the derivative of g 0 of order l by g (l) 0. Let b g (l) n denote an M-type regression spline estimator of g (l) 0 based on a training sample of size n. Under appropriate regularity conditions, it is shown that the proposed estimator, b g (l) n , achieves the optimal rate, n ?(r?l)=(2r+1) (0 l < r), of convergence of estimators for nonparametric regression when the spline knots are deterministically given.