In this paper we consider the problem of testing the equality of two multivariate distributions based on geometric graphs, constructed using the inter-point distances between the observations. These include the test based on the minimum spanning tree and the K-nearest neighbor (NN) graphs, among others. These tests are asymptotically distribution-free, universally consistent, and computationally efficient, making them particularly useful in modern applications. However, very little is known about the power properties of these tests. In this paper, using theory of stabilizing geometric graphs, we derive the asymptotic distribution of these tests under general alternatives, in the Poissonized setting. Using this, the detection threshold and the limiting local power of the test based on the K-NN graph are obtained, where interesting exponents depending on dimension emerge. This provides a way to compare and justify the performance of these tests in different examples.

中文翻译：

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基于几何图的两样本检验的渐近分布和检测阈值

在本文中，我们考虑基于几何图形测试两个多元分布是否相等的问题，几何图形使用观测值之间的点间距离构建。其中包括基于最小生成树和 K 近邻 (NN) 图等的测试。这些测试渐近无分布、普遍一致且计算效率高，这使得它们在现代应用中特别有用。然而，人们对这些测试的功效特性知之甚少。在本文中，使用稳定几何图的理论，我们在 Poissonized 设置中推导出这些测试在一般替代方案下的渐近分布。使用此，获得基于K-NN图的测试的检测阈值和限制局部功率，出现取决于维度的有趣指数。这提供了一种比较和证明这些测试在不同示例中的性能的方法。