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:我们比较了皮尔逊的线性关联系数所得出的两个随机变量之间的一致度量,这些变量的变异是随所谓的和谐-引导分布而变化的。 如此转变等级相关性的类别包括斯皮尔曼的正弦、 布隆基维斯特的贝塔和范德瓦尔登的系数。 当只需要标准一致度量的一致度量值时, 并不总能清楚哪一种转变等级相关性最适于使用。 为了解决这个问题, 我们比较了两个随机变量之间的一致度量值, 以其最佳和最差的平衡度差异来表示, 从而让某些卡通性估计者对一组依赖结构的平衡度差。 从这个方法中得出的一个简单标准是, 与小四秒的正弦相匹配性分布更为可取。 特别是, 我们显示, 布洛姆基斯特的贝特的正弦是这个意义上的最佳转变等级相关性, 而斯佩尔曼的正弦的正弦值则比范德韦登的系数更合适。 此外, 我们发现肯德尔的 tau, 尽管它不是该性质上的一个转变的等级相关性, 但是, 与Bloqm 分享了某种最佳结构的最佳结构。</s>