Measures of dependency

Although the copula is an object that summarizes completely the dependence structure of any random vector, it is an infinite dimensional object and the interpretation of its properties can be difficult when the dimension gets high. Therefore, the literature has come up with some quantification of the dependence structure that might be used as univariate summaries, of course imperfect, of certain properties of the copula at hand.

Kendall's Tau

Definition (Kendall' τ): For a copula $C$ with a density $c$, Kendall's τ is defined as:

\[\tau = 4 \int C(\bm u) \, c(\bm u) \;d\bm u -1\]

Kendall's tau can be obtained through τ(C::Copula). Its value only depends on the dependence structure and not the marginals.

Spearman's Rho

Definition (Spearman's ρ): For a copula $C$ with a density $c$, the Spearman's ρ is defined as:

\[\rho = 12 \int C(\bm u) d\bm u -3.\]

Spearman's Rho can be obtained through ρ(C::Copula). Its value only depends on the dependence structure and not the marginals.

Tail dependency

Many people are interested in the tail behavior of their dependence structures. Tail coefficients summarize this tail behavior.

Definition (Tail dependency): For a copula $C$, we define (when they exist):

\[ \begin{align} \lambda &= \lim\limits_{u \to 1} \frac{1 - 2u - C(u,..,u)}{1- u} \in [0,1]\\ \chi(u) &= \frac{2 \ln(1-u)}{\ln(1-2u-C(u,...,u))} -1\\ \chi &= \lim\limits_{u \to 1} \chi(u) \in [-1,1] \end{align}\]

When $\lambda > 0$, we say that there is a strong upper tail dependency, and $\chi = 1$. When $\lambda = 0$, we say that there is no strong upper tail dependency, and if furthermore $\chi \neq 0$ we say that there is weak upper tail dependency.

The graph of $u \to \chi(u)$ over $[\frac{1}{2},1]$ is an interesting tool to assess the existence and strength of the tail dependency. The same kind of tools can be constructed for the lower tail.

All these coefficients quantify the behavior of the dependence structure, generally or in the extremes, and are therefore widely used in the literature either as verification tools to assess the quality of fits, or even as parameters. Many parametric copulas families have simple surjections, injections, or even bijections between these coefficients and their parametrization, allowing matching procedures of estimation (a lot like moments matching algorithm for fitting standard random variables).