Robust estimation of smooth regression and spread functions and their derivatives

We consider the application of kernel weighted local polynomial regression methods to estimate regression and spread functions and their derivatives. In particular, we consider both an extension of the regression quantite methodology introduced by Koenker and Bassett (1978) and an approach based on M-estimation for heteroscedastic regression models. The present work is partly motivated by the paper of Ruppert and Wand (1994) who show that, by analysing local polynomial fitting directly as a weighted regression method rather than as an approximate kernel smooth, asymptotic results for estimating the regression function can be obtained for complex problems including vector covariates, general polynomials, derivative estimation and boundary problems. We extend their results to allow for robust fitting, for modelling general heteroscedasticity and for derivative estimation in the multivariatc case. Our results confirm that local polynomial fitting procedures produce robust estimators of the regression and spread functions and their derivatives. Moreover, it is shown that we can reduce the bias of the estimators by increasing the order of the polynomials being fitted. The excellent edge-effect behaviour of local polynomial methods extends to derivative estimation and the multivariate case. We apply the methodology to two data sets to illustrate its practical utility.