Measures of concordance and testing of independence in multivariate structure

Two random variables are concordant if one variable is large and then the other one tends to be large. Spearman’s rank correlation and Kendall’s tau can be used to measure the trend of both variables rising and falling simultaneously. For a multivariate case, most studies are based on average Spearman’s rank correlation or average Kendall’s tau, which compute bivariate measures of concordance for all pairs of variables and then average the results. A new measure of concordance which considers all the random variables simultaneously is proposed in this paper. The distribution and other relevant properties of this statistic are deduced. Since it is a U-statistic, this statistic follows an asymptotically normal distribution. Furthermore, a nonparametric test method for the independence of multivariate random variables is proposed.