Similar to all other copulas, Archimedean copulas are of the form
However, more specifically, an Archimedean is defined as
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
Is continuous and decreasing. This means that it maps a value \(x\) from \([0, \infty] \rightarrow [0, 1]\).
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.
Archimedean Copulas