On the specification of multivariate association measures and their behaviour with increasing dimension (Q2222230)

The authors consider the multivariate association measures which discover the tendency of the components of a \(d\)-variate random vector \(X = (X_1,\ldots, X_d)\) to simultaneously take large or small values. The interest is to elaborate on the generalization of bivariate association measures, namely Spearman's rho, Kendall's tau, Blomqvist's beta and Gini's gamma, for a general dimension \(d\geq 2\). Desirable properties and axioms for such generalizations are discussed. Existing generalizations are evaluated with respect to the axiom set. For a \(d\)-variate Gini's gamma, a simplified formula is developed, making its analytical computation easier. For Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension \(d\) increases is studied. Nonparametric estimation of the considered generalizations is reviewed and a nonparametric estimator of the multivariate Gini's gamma is introduced. The practical use of multivariate association measures is illustrated on a real data example.