Note on the Measures of Dependence in Terms of Copulas

The dependence structure among each risk factors has been an important topic for researches both from theoretical and applied standpoints. To measure such dependence, several characteristic quantities have been already introduced and widely employed, which include, for instance, the population version of Kendall's tau (τ) and/or Spearman's rho (ρ). Copulas, on the other hand, are well known tools for understanding the dependence relation among random variables, and the above τ and ρ are expressed in terms of copulas. In this note, we generalize these expressions. We also compute the extended formula for the Archimedean copulas as well as its generalized copulas, and pursue the possibility of its applications.