Table 1.

Three ordinary families of Archimedean Copulas (Clayton, Frank, and Gumbel Copula) and their generator, parameter space, and their formula

Family	Generator	Parameter	Formula
Clayton	${\varphi }^{\mathit{Cl}}(x)=\frac{1}{\theta }(t^{-\theta }-1)$	$-1\le \theta$	$C_{\theta }(u,v)=max{[(u^{-\theta }+v^{-\theta }-1),0]}^{-\frac{1}{\theta }}$
Frank	${\varphi }^{\mathit{Fr}}(t)=-ln\left\{\frac{{\mathrm{e}}^{-\theta t}-1}{{\mathrm{e}}^{-\theta }-1}\right\}$	$-\infty <\theta <\infty$	$C_{\theta }(u,v)=\frac{1}{\theta }ln(1+\frac{({\mathrm{e}}^{-\theta u}-1)({\mathrm{e}}^{-\theta v})}{{\mathrm{e}}^{-\theta }-1})$
Gumbel	$\varphi (t)={(-lnt)}^{\theta }$	$1\le \theta$	$C_{\theta }(u,v)={\mathrm{e}}^{-{((-ln{(u)}^{\theta })+(-ln{(v)}^{\theta }))}^{\frac{1}{\theta }}}$

$\theta$ is the parameter of the Copula called the dependence parameter, which measures the dependence between the marginal