Ridging Methods in Local Polynomial Regression

B. Seifert Th. Gasser University of Zurich Department of Biostatistics Sumatrastrasse 30 CH{8006 Z urich, Switzerland Abstract When estimating a regression function r or its th derivative, local polynomials are an attractive choice due to their exibility and asymptotic performance. Seifert & Gasser (1996) proposed local polynomial ridging to overcome problems of local polynomials with variance for random design while keeping their advantages. In this paper we present a data{adaptive spatial choice of the ridge parameter which outperforms our previously used rule of thumb. The main message is, however, that ridging is a powerful tool for the improvement of local polynomials, whereas the choice of the ridge parameter is not so decisive, but can improve the t to some extent. Local polynomial ridging is related to methods of shrinking the estimator towards the origin, to adaptive order choice, and to polynomial mixing. Relations between these methods and ridging will also be discussed. The concept of ridging is not restricted to local modi cations of estimators. Many penalized estimators like spline smoothers are global ridge estimators. A penalized local polynomial estimator (Seifert & Turlach 1996) shows an attractive performance and shares all asymptotic properties with local polynomials.