Sums of Functions of Nearest Neighbor Distances, Moment Bounds, Limit Theorems and a Goodness of Fit Test

Abstract

We study the limiting behavior of sums of functions of nearest neighbor distances for an $m$ dimensional sample. We establish a central limit theorem and moment bounds for such sums and an invariance principle for the empirical process of nearest neighbor distances. As a consequence we obtain the asymptotic behavior of a practicable goodness of fit test based on nearest neighbor distances.