Handbook of Heavy Tailed Distributions in Finance

3.4 Kendall’s tau and Spearman’s rho

In this section we discuss two important measures of dependence (concordance) known as Kendall's tau and Spearman's rho. They provide the perhaps best alternatives to the linear correlation coefficient as a measure of dependence for nonelliptical distributions, for which the linear correlation coefficient is inappropriate and often misleading. For more details about Kendall's tau and Spearman's rho and their estimators (sample versions) we refer to Kendall and Stuart (1979), Kruskal (1958), Lehmann (1975), Capéraà and Genest (1993). For other interesting scalar measures of dependence see Schweizer and Wolff (1981).

Definition 3.4. Kendall's tau for the random vector ${(X,Y)}^T$ is defined as

$\tau \left(X,Y\right)=\mathbb{P}\{\left(X-\tilde{X}\right)(Y-\stackrel{}{Y}$