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 ${ℝ}^d$ and on lower-dimensional manifolds embedded in ${ℝ}^d$ 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 $\alpha$-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests. The tests are applied to two data sets of gamma-ray bursts in astronomy.