A Weighted k-Nearest Neighbor Density Estimate for Geometric Inference

Abstract Motivated by a broad range of potential applications in topological and geometric inference, we introduce a weighted version of the knearest neighbor density estimate. Various pointwise consistency results of this estimate are established. We present a general central limit theorem under the lightest possible conditions. In addition, a strong approximation result is obtained and the choice of the optimal set of weights is discussed. In particular, the classical k-nearest neighbor estimate is not optimal in a sense described in the manuscript. The proposed method has been implemented to recover level sets in both simulated and real-life data.