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.
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机译：我们基于$ k $加权卷的幂的总和，给出了$ \ mathbb {R} ^ d $和嵌入在$ \ mathbb {R} ^ d $中的低维流形上的拟合优度测试的统一方法。 -th最近的相邻球体。我们证明了在原假设和固定替代条件下一类检验统计量的渐近正态性。在这样的选择下，检验统计量的缩放版本收敛到概率分布之间的$ \α$熵。仿真研究表明，该程序是建立拟合优度测试的重要竞争者。
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