基于投影的两样本分布相等的非参数检验

Nonparametric two-sample tests for equality of distributions using projections

本文通过利用投影推广经典的Cramér-von Mises度量,研究能够适应于高维数据的两样本分布相等的非参数检验.本文所提出的新度量是非负的,且新度量等于零当且仅当两总体具有相同分布,这确保相应的检验能够探测任意的备择假设.本文所推荐的检验统计量具有精确的代数表示,不依赖于任何冗余参数.由于新度量的定义不需要任何的矩条件,本文所提出的检验方法能够对数据中强影响值和异常值稳健.在传统的“大样本、固定维数”和新型的“固定样本、高维数”框架下,研究了新检验统计量的渐近性质.在传统的“大样本、固定维数”框架下,证明所推荐的检验功效不依赖于两样本量的比值的大小,这确保新检验可以适用于非平衡样本的数据.在新型的“固定样本、高维数”框架下,证明所推荐的检验功效主要由两总体的位置和尺度差异决定.在这一框架下,本文进一步修正所推荐的检验统计量使其在探测两总体的位置和尺度差异时具有更高的功效.数值研究表明本文所提出的检验是有效和切实可行的.

We propose a nonparametric two-sample test,which generalizes the Cramér-von Mises test through projections,to test for equality of two distributions in high dimensions.The population version of our proposed generalized Cramér-von Mises statistic is nonnegative and equals zero if and only if the two distributions are identical,ensuring that our proposed test is consistent against all the fixed alternatives.In addition,our proposed test statistic has an explicit form and is completely free of tuning parameters.It requires no moment conditions and hence is robust to the presence of outliers and heavy-tail observations.We study the asymptotic behaviors of our proposed test under both the“large sample size,fixed dimension”and the“fixed sample size,large dimension”paradigms.In the former paradigm,we show that the asymptotic power of our proposed test does not depend on the size ratio of the two random samples.This ensures that our proposed test can be readily applied to imbalanced samples.In the latter paradigm,we observe that,surprisingly,the two distributions are equal if and only if their first two moments are equal.In this paradigm,we suggest tailoring our proposed test to detect location shifts and scale differences,which further enhances the power performance of our proposed test significantly.Numerical studies confirm that our proposals are superior to many existing tests in high-dimensional two-sample test problems.