Title: Power properties of the two-sample test based on the nearest neighbors graph Show Chinese title

Title: 基于最近邻图的两样本检验的功率特性

Abstract: In this paper, we study the problem of testing the equality of two multivariate distributions. One class of tests used for this purpose utilizes geometric graphs constructed using inter-point distances. So far, the asymptotic theory of these tests applies only to graphs which fall under the stabilizing graphs framework of \citet{penroseyukich2003weaklaws}. We study the case of the $K$-nearest neighbors graph where $K=k_N$ increases with the sample size, which does not fall under the stabilizing graphs framework. Our main result gives detection thresholds for this test in parametrized families when $k_N = o(N^{1/4})$, thus extending the family of graphs where the theoretical behavior is known. We propose a 2-sided version of the test which removes an exponent gap that plagues the 1-sided test. Our result also shows that increasing the number of nearest neighbors boosts the power of the test. This provides theoretical justification for using denser graphs in testing equality of two distributions. Show Chinese abstract

Abstract: 本文研究了检验两个多元分布是否相等的问题。一类用于此目的的检验方法利用了基于点间距离构造的几何图。到目前为止，这些检验的渐近理论仅适用于属于\citet{penroseyukich2003weaklaws}的稳定图框架的图。我们研究了当$K=k_N$随样本量增加时的$K$-最近邻图的情况，这并不属于稳定图框架。我们的主要结果给出了当$k_N = o(N^{1/4})$时，该检验在参数化族中的检测阈值，从而扩展了已知理论行为的图类。我们提出了一个两面版本的检验，消除了困扰单面检验的指数差距。我们的结果还表明，增加最近邻的数量可以提高检验的功效。这为在检验两个分布相等时使用更密集的图提供了理论依据。