# Archimedean Copulas¶

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.