Comparison of correlation-based measures of concordance in terms of asymptotic variance

TL;DR

This paper compares various measures of concordance based on their asymptotic variances, identifying Blomqvist's beta as optimal among transformed rank correlations and highlighting the performance differences of Spearman's rho and van der Waerden's coefficient.

Contribution

It introduces a criterion based on asymptotic variance to evaluate and compare measures of concordance, revealing the optimality of Blomqvist's beta and the relative performance of other measures.

Findings

Blomqvist's beta has the smallest asymptotic variance among the measures studied.

Spearman's rho outperforms van der Waerden's coefficient in the comparison.

Kendall's tau shares an optimal structure with Blomqvist's beta.

Abstract

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.