zbMATH — the first resource for mathematics

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.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference