An unbiased rank-based estimator of the Mann–Whitney variance including the case of ties

Many estimators of the variance of the well-known unbiased and uniform most powerful estimator of the Mann–Whitney effect, are considered in the literature. Some of these estimators are only valid in cases of no ties or are biased in small sample sizes where the amount of bias is not discussed. Here, we derive an unbiased estimator based on different rankings, the so-called ’placements’ (Orban and Wolfe in Commun Stat Theory Methods 9:883–904, 1980), which is therefore easy to compute. This estimator does not require the assumption of continuous distribution functions 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 distribution functions 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 confidence interval do not only depend on the quality of the approximation of by a normal distribution but also an additional unknown bias caused by the variance estimator. Finally, it is shown that this estimator is L2-consistent.