While there appears to be a general consensus in the literature on the definition of the estimand and estimator associated with the Wilcoxon-Mann-Whitney test, it seems somewhat less clear as to how best to estimate the variance. In addition to the Wilcoxon-Mann-Whitney test, we review different proposals of variance estimators consistent under both the null hypothesis and the alternative. Moreover, in case of small sample sizes, an approximation of the distribution of the test statistic based on the t-distribution, a logit transformation and a permutation approach have been proposed. Focussing as well on different estimators of the degrees of freedom as regards the t-approximation, we carried out simulations for a range of scenarios, with results indicating that the performance of different variance estimators in terms of controlling the type I error rate largely depends on the heteroskedasticity pattern and the sample size allocation ratio, not on the specific type of distributions employed. By and large, a particular t-approximation together with Perme and Manevski's variance estimator best maintains the nominal significance level

中文翻译：

---

小样本中的非参数 Behrens-Fisher 问题

虽然文献中似乎对与 Wilcoxon-Mann-Whitney 检验相关的估计量和估计量的定义达成了普遍共识，但对于如何最好地估计方差似乎不太清楚。除了 Wilcoxon-Mann-Whitney 检验之外，我们还回顾了在原假设和备择假设下一致的方差估计量的不同提议。此外，在小样本量的情况下，已经提出了基于 t 分布的检验统计量分布近似值、logit 变换和置换方法。还关注关于 t 近似的自由度的不同估计量，我们对一系列场景进行了模拟，结果表明，不同方差估计器在控制 I 类错误率方面的性能在很大程度上取决于异方差模式和样本量分配比，而不是取决于所采用的特定分布类型。总的来说，特定的 t 近似值与 Perme 和 Manevski 的方差估计量一起最好地保持名义显着性水平