On the specification of multivariate association measures and their behaviour with increasing dimension
Abstract
In this paper 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 >= 2. Desirable properties and axioms for such generalizations are discussed, where special attention is given to the impact of the addition of: (i) an independent random variable to a random vector; (ii) a conical combination of all components; (iii) a set of arbitrary random components.
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.
Further, for Archimedean and meta-elliptical copulas the asymptotic behaviour when the dimension d increases is studied. Nonparametric estimation of the considered generalizations of multivariate association measures 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. (C) 2020 Elsevier Inc. All rights reserved.