Estimation and Comparison of Correlation-based Measures of Concordance

We address the problem of estimating and comparing 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 distribution. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. To answer which transformed rank correlation is best to use, we propose to compare them in terms of their best and worst asymptotic variances on a given set of copulas. A criterion derived from this approach is that concordance-inducing distributions with smaller variances of squared random variables are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau also attains the optimal asymptotic variances that Blomqvist's beta does, although Kendall's tau is not a transformed rank correlation.