# 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.

Archimedean Copulas