Testing of Multivariate Concordance and Pitman Asymptotic Relative Efficiency

Measuring the concordance among multiple variables is a frequent challenge in practical applications. Several well-established methods for measuring concordance exist for the binary case, such as Spearman’s rank correlation coefficient, Kendall’s rank correlation coefficient, and Blomqvist’s beta coefficient. In the multivariate case, most existing measures are based on pairwise averages, such as the average Spearman’s rank correlation coefficient and the average Kendall’s tau rank correlation coefficient. These measures of concordance primarily assess pairwise concordance and do not capture the overall characteristics of multiple variables, leading to issues with unexplained negative concordance. This paper introduces a new measure of concordance based on the overall structure of multiple variables. It also investigates the properties of relevant statistics. Based on this, a non-parametric test method for the concordance of multivariate data is proposed. Simulation results indicate that, in terms of Pitman asymptotic relative efficiency, the proposed test method outperforms that of Deng 1.