Variable bandwidth in nonparametric regression

In the context of estimating a probability density function, use of a suitable variable bandwidth is known to improve the rate of convergence of the resulting kernel density estimator. In this paper we show that the same kind of improvement is possible in the regression setting. In particular, we find that the fast rate of convergence derived by Hall (1990), using a bandwidth variation method that depends on the underlying regression function, still holds when one uses an estimate of the regression function as a pilot.