Statistica Sinica 5(1995), 303-318

GLOBAL CONVERGENCE RATES OF B-SPLINE

M-ESTIMATORS IN NONPARAMETRIC REGRESSION

Peide Shi and Guoying Li

Academia Sinica

Abstract: 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, g₀, 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 g₀ is smoothed up to order r>1/2 and denote the derivative of g₀ of order l by Let denote an M-type regression spline estimator of based on a training sample of size n. Under appropriate regularity conditions, it is shown that the proposed estimator, ,achieves the optimal rate,n^{-(r-1)/(2r+1)}(0≤ l < r), of convergence of estimators for nonparametric regression when the spline knots are deterministically given.

Key words and phrases: B-spline function, M-estimator, nonparametric regression, optimal rate of convergence.