Rates of convergence for the k-nearest neighbor estimators with smoother regression functions

Let (X, Y) be a R d × R - valued random vector. In regression analysis one wants to estimate the regression function m ( x ) ≔ E ( Y | X = x ) from a data set. In this paper we consider the rate of convergence for the k-nearest neighbor estimators in case that X is uniformly distributed on [ 0,1 ] d , Var ( Y | X = x ) is bounded, and m is (p, C)-smooth. It is an open problem whether the optimal rate can be achieved by a k-nearest neighbor estimator for 1 < p ≤ 1.5 . We solve the problem affirmatively. This is the main result of this paper. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L2 error.