Archimedean Copulas

Similar to all other copulas, Archimedean copulas are of the form

\begin{align*} H(\textbf{x}) = C(F_1(x_1), \dots, F_d(x_d)) \quad \textbf{x} \in \mathbb{R}^d \end{align*}

However, more specifically, an Archimedean is defined as

\begin{align*} C(\textbf{u}) = \psi(\psi^{-1}(u_1), \dots, \psi^{-1}(u_d)) \quad \textbf{u} \in [0, 1]^d \end{align*}

So in this class of copulas, you would first need a generator function, defined as \(\psi\). This function has the nice property that it is defined by a single value \(\theta\). Thus when you fit an Archimedean copula, you only need to “learn” this unknow value \(\theta\).

In general, to be an Archimedean generator, \(\psi\) must be a function that

  1. Is continuous and decreasing. This means that it maps a value \(x\) from \([0, \infty] \rightarrow [0, 1]\).

  2. Has derivatives up to \(k = d-2\) where \(d\) is the dimension of the copula and that \((-1)^k\psi^k(x) \geq 0 \quad \forall k \in \{0, \dots, d-2\}, x \in (0, \infty)\)

One such function is the exponential function. In fact, if we let \(\psi(x) = \exp(-x), t \in [0, \infty]\), we would get the Gumbel copula.