Adaptively weighted kernel regression

We develop a new kernel-based local polynomial methodology for nonparametric regression based on optimising a linear combination of several loss functions. Optimal weights for least squares and quantile loss functions can be chosen to provide maximum efficiency and these optimal weights can be estimated from data. The resulting estimators are at least as efficient as those provided by existing procedures, but can be much more efficient for many distributions. The data-based weights adapt to the tails of the error distribution resulting in a procedure which is both robust and resistant. Furthermore, the assumption of homogeneous error variance is not required. To illustrate its practical use, we apply the proposed method to model the motorcycle data.