Asymptotic Normality of Nearest Neighbor Regression Function Estimates

Let (X, Y) be a random vector in the plane. We show that a smoothed N. N. estimate of the regression function m (x) = E (Y X = x) is asymptotically normal under conditions much weaker than needed for the Nadaraya-Watson estimate. It also turns out that N. N. estimates are more efficient than kernel-type estimates if (in the mean) there are few observations in neighborhoods of x.