A New Approach to the Nonparametric Behrens-Fisher Problem with Compatible Confidence Intervals

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.