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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 ofpowers of weighted volumes of $k$-th nearest neighbor spheres. We proveasymptotic normality of a class of test statistics under the null hypothesisand under fixed alternatives. Under such alternatives, scaled versions of thetest statistics converge to the $\alpha$-entropy between probabilitydistributions. A simulation study shows that the procedures are seriouscompetitors to established goodness-of-fit tests.

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