Variable bandwidth and One - step Local

We study a robust version of local linear regression smoothers augmented with variable bandwidth. The proposed method inherits the advantages of local polynomial regression and overcomes lack of robustness of least-squares techniques. The use of variable bandwidth enhances the exibility of the resulting local M-estimators and makes them possible to cope well with spatially inhomogeneous curves, heteroscedastic errors and nonuniform design densities. Under appropriate regularity conditions, it is shown that the proposed estimators exist and are asymptotically normal. Based on the robust estimation equation, we introduce one-step local M-estimators to reduce computational burden. It is demonstrated that the one-step local M-estimators share the same asymptotic distributions as the fully iterative M-estimators, as long as the initial estimators are good enough. In other words, the one-step local M-estimators reduce signiicantly the computation cost of the fully iterative M-estimators without deteriorating their performance. This fact is also illustrated via simulations.