Robust Non-parametric Function Estimation

The bias of kernel methods based on local constant fits can have an adverse effect when the derivative of the marginal density or that of the regression function is large. The drawback can be repaired by considering a class of kernel estimators based on local linear fits. These estimators have the desired asymptotic properties and can be used to estimate conditional quantiles and to robustify the usual mean regression. The conditional asymptotic normality of these estimators at both boundary and interior points is established. An important consequence of the study is that the proposed method has the desired sampling properties at both boundary and interior points of the support of the design density. Therefore, our procedure does not require boundary modifications. Applications of such a local linear approximation method are discussed.