Many estimators of the variance of the well-known unbiased and uniform most powerful estimator $\htheta$ of the Mann-Whitney effect, $\theta = P(X < Y) + \nfrac12 P(X=Y)$, are considered in the literature. Some of these estimators are only valid in case of no ties or are biased in case of small sample sizes where the amount of the bias is not discussed. Here we derive an unbiased estimator that is based on different rankings, the so-called 'placements' (Orban and Wolfe, 1980), and is therefore easy to compute. This estimator does not require the assumption of continuous \dfs\ and is also valid in the case of ties. Moreover, it is shown that this estimator is non-negative and has a sharp upper bound which may be considered an empirical version of the well-known Birnbaum-Klose inequality. The derivation of this estimator provides an option to compute the biases of some commonly used estimators in the literature. Simulations demonstrate that, for small sample sizes, the biases of these estimators depend on the underlying \dfs\ and thus are not under control. This means that in the case of a biased estimator, simulation results for the type-I error of a test or the coverage probability of a \ci\ do not only depend on the quality of the approximation of $\htheta$ by a normal \db\ but also an additional unknown bias caused by the variance estimator. Finally, it is shown that this estimator is $L_2$-consistent.

翻译：文献中已提出多种用于估计Mann-Whitney效应$\theta = P(X < Y) + \nfrac12 P(X=Y)$的经典无偏一致最有效估计量$\htheta$的方差估计量。其中部分估计量仅适用于无结值情形，或在样本量较小时存在偏差，且偏差程度尚未得到充分讨论。本文基于不同排序方法（即所谓"位置统计量"，Orban与Wolfe，1980）推导出一种计算简便的无偏估计量。该估计量不要求分布函数连续的假设，在存在结值的情况下依然有效。进一步证明该估计量具有非负性，且存在明确上界，可视为经典Birnbaum-Klose不等式的经验版本。通过推导该估计量，为计算文献中常用估计量的偏差提供了新途径。模拟实验表明，在小样本情形下，这些估计量的偏差取决于潜在分布函数，因而具有不可控性。这意味着当使用有偏估计量时，假设检验的第一类错误率或置信区间的覆盖概率不仅受$\htheta$正态逼近精度的影响，还会受到方差估计量引入的未知偏差干扰。最后，本文证明该估计量具有$L_2$相合性。