Handbook of Heavy Tailed Distributions in Finance

6.3 Kendall’s tau revisited

Recall that Kendall's tau for a copula $C$ can be expressed as a double integral of $C$. This double integral is in most cases not straightforward to evaluate. However for an Archimedean copula, Kendall's tau can be expressed as an (one-dimensional) integral of the generator and its derivative, as shown in the following theorem from Genest and MacKay (1986a).

Theorem 6.5. Let $X$ and $Y$ be random variables with Archimedean copula ...