Interpoint distance based two sample tests in high dimension
Interpoint distance based two sample tests in high dimension
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DOI：
10.3150/20-bej1270
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发表时间：
2019-02
期刊：
影响因子：
1.5
通讯作者：
Changbo Zhu;Xiaofeng Shao
中科院分区：
文献类型：
--
作者：
Changbo Zhu;Xiaofeng Shao
本文研究了一类基于点间距离的高维低样本量条件下的两样本检验统计量。我们的检验统计量包括众所周知的与高斯核和拉普拉斯核的能量距离和最大平均差异，并通过置换获得临界值。我们表明，当两个高维分布对应于相同的边际分布而在分布的其他方面不同时，所有这些检验都是不一致的。基于能量距离和最大均值差异的检验主要针对边际均值和方差之间的差异，而基于$L^1$-距离的检验可以捕获边际分布的差异。我们的理论揭示了基于点间距离的测试的局限性，不同距离度量的影响，以及高维排列测试的行为。并给出了一些仿真结果和一个实际数据说明来证实我们的理论发现。
In this paper, we study a class of two sample test statistics based on inter-point distances in the high dimensional and low sample size setting. Our test statistics include the well-known energy distance and maximum mean discrepancy with Gaussian and Laplacian kernels, and the critical values are obtained via permutations. We show that all these tests are inconsistent when the two high dimensional distributions correspond to the same marginal distributions but differ in other aspects of the distributions. The tests based on energy distance and maximum mean discrepancy are mainly targeting the differences between marginal means and variances, whereas the test based on $L^1$-distance can capture the difference in marginal distributions. Our theory sheds new light on the limitation of inter-point distance based tests, the impact of different distance metrics, and the behavior of permutation tests in high dimension. Some simulation results and a real data illustration are also presented to corroborate our theoretical findings.