Distribution-free consistency of a nonparametric kernel regression estimate and classification

It is shown that the kernel estimate of the regressionE(Y|X = x)is weakly or strongly consistent for almost allx(\mu), where\muis the probability measure ofX. The result is valid for any distribution ofX. The asymptotical optimality of classification rules derived from the estimate is examined. The optimality is independent of class distributions, i.e., it is distribution-free.