We propose a new test to address the nonparametric Behrens-Fisher problem involving different distribution functions in the two samples. Our procedure tests the null hypothesis $\mathcal{H}_0: \theta = \frac{1}{2}$, where $\theta = P(X<Y) + \frac{1}{2}P(X=Y)$ denotes the Mann-Whitney effect. No restrictions on the underlying distributions of the data are imposed with the trivial exception of one-point distributions. The method is based on evaluating the ratio of the variance $\sigma_N^2$ of the Mann-Whitney effect estimator $\widehat{\theta}$ to its theoretical maximum, as derived from the Birnbaum-Klose inequality. Through simulations, we demonstrate that the proposed test effectively controls the type-I error rate under various conditions, including small sample sizes, unbalanced designs, and different data-generating mechanisms. Notably, it provides better control of the type-1 error rate compared to the widely used Brunner-Munzel test, particularly at small significance levels such as $\alpha \in \{0.01, 0.005\}$. Additionally, we derive range-preserving compatible confidence intervals, showing that they offer improved coverage over those compatible to the Brunner-Munzel test. Finally, we illustrate the application of our method in a clinical trial example.

翻译：本文针对两样本分布函数不同的非参数Behrens-Fisher问题提出了一种新的检验方法。我们的程序检验原假设$\mathcal{H}_0: \theta = \frac{1}{2}$，其中$\theta = P(X<Y) + \frac{1}{2}P(X=Y)$表示Mann-Whitney效应。除单点分布外，该方法不对数据的底层分布施加任何限制。该方法基于评估Mann-Whitney效应估计量$\widehat{\theta}$的方差$\sigma_N^2$与其理论最大值的比值，该理论最大值由Birnbaum-Klose不等式导出。通过模拟实验，我们证明所提出的检验方法在各种条件下（包括小样本量、不平衡设计及不同数据生成机制）均能有效控制第一类错误率。值得注意的是，与广泛使用的Brunner-Munzel检验相比，该方法在较小显著性水平（如$\alpha \in \{0.01, 0.005\}$）下能更好地控制第一类错误率。此外，我们推导了保范围的相容置信区间，结果表明其覆盖性能优于与Brunner-Munzel检验相容的置信区间。最后，我们通过临床试验案例展示了该方法的应用。