On the Strong Universal Consistency of Nearest Neighbor Regression Function Estimates

Authors: Luc Devroye, László Györfi, Adam Krzyżak, Gábor Lugosi - The Annals of Statistics 1994 cited by 305

Two results are presented concerning the consistency of the $k$-nearest neighbor regression estimate. We show that all modes of convergence in $L_1$ (in probability, almost sure, complete) are equivalent if the regression variable is bounded. Under the additional conditional $k/\log n \rightarrow \infty$ we also obtain the strong universal consistency of the estimate.