We consider the Mann–Whitney test for two‐phase stratified sampling. In this design, the independent and identically distributed (i.i.d.) sample is obtained at the first phase and then stratified based on auxiliary variables. At the second phase, stratified subsamples are obtained without replacement to collect variables of interest. The resultant data are biased and dependent sample due to stratification and sampling without replacement. This setting is different from the one considered by the existing method called the van Elteren test that treats multiple i.i.d. samples from nonoverlapping subsets of the infinite population. We propose the inverse probability weighted Mann–Whitney statistic and study its asymptotic properties. The proposed test is shown to have a correct size and power 1 in the limit. Simulation study is presented to illustrate the finite sample performance of our test.

中文翻译：

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曼恩-惠特尼检验用于两阶段分层抽样

我们考虑使用Mann-Whitney检验进行两阶段分层抽样。在此设计中，在第一阶段获得独立且分布均匀的（iid）样本，然后根据辅助变量进行分层。在第二阶段，无需分层即可获得分层的子样本，以收集感兴趣的变量。由于分层和采样而不替换，结果数据有偏差且依赖于样本。此设置不同于现有方法所考虑的称为van Elteren检验的设置，该方法处理来自无限总体的非重叠子集的多个iid样本。我们提出了逆概率加权的Mann-Whitney统计量，并研究了其渐近性质。建议的测试显示出正确的大小，并且极限功率为1。