A suggestion for a multivariate concordance coefficient

In the present paper we will introduce a coecient of multivariate association i.e. association in a d-variate vector of observations x = (x1; : : : ; xd), where d 2 and where each xj is itself a vector of n observations. We order the observations, divide them in slices and count how many times one observation in the r-th slice of any of the d distributions also belongs to the r-th slice of any of the others. The greater the number of overlaps between the units belonging to corresponding slices, the greater the concordance between the d distributions. This is the simple and intuitive idea our multivariate association coecient stems from. It is in fact a multidimensional concordance coecient since it assumes comonotonicity for all variables.