Liouville Copulas

Archimedean copulas have been widely used in the literature due to their nice decomposition properties and easy parametrization. The interested reader can refer to the extensive literature [24–30, 30–33] on Archimedean copulas, their nesting extensions and most importantly their estimation.

One major drawback of the Archimedean family is that these copulas have exchangeable marginals (i.e., $C(\bm u) = C(\mathrm{p}(\bm u))$ for any permutation $p(\bm u)$ of $u_1,...,u_d$): the dependence structure is symmetric, which might not be a wanted property. However, from the Radial-simplex expression, we can easily extrapolate a little and take for $\bm S$ a non-uniform distribution on the simplex.

Liouville's copulas share many properties with Archimedean copulas, but are not exchangeable anymore. This is an easy way to produce non-exchangeable dependence structures. See [15] for a practical use of this property.

Note that Dirichlet distributions are constructed as $\bm S = \frac{\bm G}{\langle \bm 1, \bm G\rangle}$, where $\bm G$ is a vector of independent Gamma distributions with unit scale (and potentially different shapes: taking all shapes equal yields the Archimedean case).