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Global convergence rates of B-spline M-estimators in nonparametric regression. (English) Zbl 0824.62035
Summary: 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 $$g_ 0^{(l)}$$. Let $$\widehat {g}_ n^{(l)}$$ denote an $$M$$-type regression spline estimator of $$g_ 0^{(l)}$$ based on a training sample of size $$n$$. Under appropriate regularity conditions, it is shown that the proposed estimator, $$\widehat {g}_ n^{(l)}$$, achieves the optimal rate, $$n^{(r- l)/ (2r+1)}$$ $$(0\leq l<r)$$, of convergence of estimators for nonparametric regression when the spline knots are deterministically given.

##### MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference