Multivariate goodness-of-fit on flat and curved spaces via nearest neighbor distances

Abstract

We present a unified approach to goodness-of-fit testing in ${\mathbb{R}}^d$ and on lower-dimensional manifolds embedded in ${\mathbb{R}}^d$ based on sums of powers of weighted volumes of $k$-th nearest neighbor spheres. We prove asymptotic normality of a class of test statistics under the null hypothesis and under fixed alternatives. Under such alternatives, scaled versions of the test statistics converge to the $\hat{I}\pm$-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests.

22 pages, 1 figure, 9 tables