Nonparametric <i>K</i>-Sample Tests via Dynamic Slicing

<i>K</i>-sample testing problems arise in many scientific applications and have attracted statisticians’ attention for many years. We propose an omnibus nonparametric method based on an optimal discretization (aka “slicing”) of continuous random variables in the test. The novelty of our approach lies in the inclusion of a term penalizing the number of slices (i.e., the resolution of the discretization) so as to regularize the corresponding likelihood-ratio test statistic. An efficient dynamic programming algorithm is developed to determine the optimal slicing scheme. Asymptotic and finite-sample properties such as power and null distribution of the resulting test statistic are studied. We compare the proposed testing method with some existing well-known methods and demonstrate its statistical power through extensive simulation studies as well as a real data example. A dynamic slicing method for the one-sample testing problem is further developed and studied under the same framework. Supplementary materials including technical derivations and proofs are available online.K样本检验（K-sample testing）问题在诸多科学应用中均有出现，多年来一直受到统计学者的广泛关注。本文针对检验中的连续随机变量，提出一种基于最优离散化（即“切片”（slicing））的综合非参数方法。本方法的新颖之处在于引入了一项对切片数量（即离散化的分辨率）进行惩罚的项，以对相应的似然比检验统计量进行正则化。本文开发了一种高效的动态规划算法，用以确定最优切片方案。针对所得检验统计量的渐近性质与有限样本性质（如检验功效与原假设分布），本文展开了系统研究。本文将所提检验方法与若干现有经典方法进行对比，并通过大规模模拟实验与一则真实数据案例验证了其统计功效。此外，本文还在同一框架下，进一步提出并研究了针对单样本检验问题的动态切片方法。包含技术推导与证明过程的补充材料可在线获取。