A multivariate version of Hoeffding's Phi-Square

AMS 2000 subject class: Primary 62H20 60F05 62G20 Secondary 60G15 62G10 Keywords: Multivariate measure of association Copula Nonparametric estimation Empirical copula process Weak convergence Nonparametric bootstrap Strong mixing a b s t r a c t A multivariate measure of association is proposed, which extends the bivariate copula-based measure Phi-Square introduced by Hoeffding [22]. We discuss its analytical properties and calculate its explicit value for some copulas of simple form; a simulation procedure to approximate its value is provided otherwise. A nonparametric estimator for multivariate Phi-Square is derived and its asymptotic behavior is established based on the weak convergence of the empirical copula process both in the case of independent observations and dependent observations from strictly stationary strong mixing sequences. The asymptotic variance of the estimator can be estimated by means of nonparametric bootstrap methods. For illustration, the theoretical results are applied to financial asset return data.