Four Measures of Association and Their Representations in Terms of Copulas

Abstract

Four measures of association, namely, Spearman’s ρ, Kendall’s τ, Blomqvist’s β and Hoeffding’s Φ2, are expressed in terms of copulas. Conveniently, this article also includes explicit expressions for their empirical counterparts. Moreover, copula representations of the four coefficients are provided for the multivariate case, and several specific applications are pointed out. Additionally, a numerical study is presented with a view to illustrating the types of relationships that each of the measures of association can detect.

Keywords

• Blomqvist’s correlation coefficient
• copulas
• Hoeffding’s dependence index
• Kendall’s rank correlation coefficient
• multivariate measures of dependence
• Spearman’s rank correlation