In this paper, we consider the problem of testing the equality of two multivariate distributions based on geometric graphs constructed using the interpoint distances between the observations. These include the tests 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 the 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.

翻译：在本文中,我们考虑了测试两个多变量分布的等值问题,这两个多变量分布基于利用观测之间的间距构造的几何图形。 其中包括基于最小横幅树和美元最近的相邻图等的测试。 这些测试是零星分布、普遍一致和计算效率的, 使得这些测试在现代应用中特别有用。 但是,对这些测试的功率特性知之甚少。 在本文中,我们使用稳定几何图理论,在普瓦森化的环境下,根据一般替代方法, 得出这些测试的无症状分布。 使用这个方法, 检测阈值和基于$- 美元最近的近邻图的局部功率, 获得的测试的局部功率因大小而异。 这为在不同例子中比较和论证这些测试的性能提供了一种方法。