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

We present a unified approach to goodness-of-fit testing in ℝd and on lower-dimensional manifolds embedded in ℝd 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 α-entropy between probability distributions. A simulation study shows that the procedures are serious competitors to established goodness-of-fit tests.