The Hilbert Kernel Regression Estimate

Let (X, Y) be an Rd×R-valued regression pair, whereXhas a density andYis bounded. Ifni.i.d. samples are drawn from this distribution, the Nadaraya–Watson kernel regression estimate in Rdwith Hilbert kernelK(x)=1/‖x‖dis shown to converge weakly for all such regression pairs. We also show that strong convergence cannot be obtained. This is particularly interesting as this regression estimate does not have a smoothing parameter.