The Lower Bound for the Nearest Neighbor Estimators with (p, C)-Smooth Regression Functions
Let (X, Y) be a R-d x R-valued random vector. In regression analysis one wants to estimate the regression function m(x) := E(Y vertical bar X = x) from a data set. In this paper we consider the convergence rate of the error for the k nearest neighbor estimators in case that m is (p, C)-smooth. It is known that the minimax rate is unachievable by any k nearest neighbor estimator for p > 1.5 and d = 1. We generalize this result to any d >= 1. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected L-2 error.
nearest neighbor regression nonparametric estimation rate of convergence