A fundamental functional in nonparametric statistics is the Mann-Whitney functional ${\theta} = P (X < Y )$ , which constitutes the basis for the most popular nonparametric procedures. The functional ${\theta}$ measures a location or stochastic tendency effect between two distributions. A limitation of ${\theta}$ is its inability to capture scale differences. If differences of this nature are to be detected, specific tests for scale or omnibus tests need to be employed. However, the latter often suffer from low power, and they do not yield interpretable effect measures. In this manuscript, we extend ${\theta}$ by additionally incorporating the recently introduced distribution overlap index (nonparametric dispersion measure) $I_2$ that can be expressed in terms of the quantile process. We derive the joint asymptotic distribution of the respective estimators of ${\theta}$ and $I_2$ and construct confidence regions. Extending the Wilcoxon- Mann-Whitney test, we introduce a new test based on the joint use of these functionals. It results in much larger consistency regions while maintaining competitive power to the rank sum test for situations in which {\theta} alone would suffice. Compared with classical omnibus tests, the simulated power is much improved. Additionally, the newly proposed inference method yields effect measures whose interpretation is surprisingly straightforward.

翻译：非参数统计中的一个基本泛函是曼-惠特尼泛函 ${\theta} = P (X < Y )$，它构成了最常用非参数方法的基础。该泛函 ${\theta}$ 衡量两个分布之间的位置或随机趋势效应。其局限性在于无法捕捉尺度差异。若需检测此类差异，则需使用特定的尺度检验或综合检验。然而，后者通常功效较低，且无法提供可解释的效应度量。本文通过额外引入近期提出的分布重叠指数（非参数离散度量）$I_2$（该指数可用分位过程表示）对 ${\theta}$ 进行了扩展。我们推导了 ${\theta}$ 和 $I_2$ 各自估计量的联合渐近分布，并构建了置信区域。通过扩展威尔科克森-曼-惠特尼检验，我们提出了一种基于这些泛函联合使用的新检验方法。该方法在仅使用 ${\theta}$ 即足够的情景中，相较于秩和检验，在保持竞争性功效的同时，获得了更大的检验一致区域。与经典综合检验相比，模拟功效显著提升。此外，新提出的推断方法所产生的效应度量，其解释具有惊人的直观性。