We compare measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distributions. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient. When only the standard axioms of measures of concordance are required, it is not always clear which transformed rank correlation is most suitable to use. To address this question, we compare measures of concordance in terms of their best and worst asymptotic variances of some canonical estimators over a certain set of dependence structures. A simple criterion derived from this approach is that concordance-inducing distributions with smaller fourth moment are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation in this sense, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau, although it is not a transformed rank correlation of that nature, shares a certain optimal structure with Blomqvist's beta.

翻译：我们比较了通过将两个随机变量变换为服从所谓“一致性诱导分布”后计算其皮尔逊线性相关系数得到的一致性测度。这类变换秩相关包括斯皮尔曼ρ、布洛姆奎斯特β和范德瓦尔登系数。当仅需满足一致性测度的标准公理时，究竟哪种变换秩相关最为适用并不明确。针对这一问题，我们在特定依赖结构集合上，从某些典型估计量的最优与最劣渐近方差角度比较了这些一致性测度。基于该方法导出的简单准则是：具有更小四阶矩的一致诱导分布更优。特别地，我们证明了布洛姆奎斯特β在该意义下是最优变换秩相关，而斯皮尔曼ρ优于范德瓦尔登系数。此外，我们发现肯德尔τ虽非此类变换秩相关，却与布洛姆奎斯特β共享某种最优结构。