Frank (Archimedean) $$\frac{-1}{\uptheta}\ln \left(1+\frac{\left({\mathrm{e}}^{-\uptheta \mathrm{u}}-1\right)\left({\mathrm{e}}^{-\uptheta \mathrm{v}}-1\right)}{{\mathrm{e}}^{-\uptheta}-1}\right)$$ θ ≠ 0
Gumbel Hougaard (Archimedean) $${\mathrm{e}}^{{\left[{\left(-\ln \mathrm{u}\right)}^{\uptheta}+{\left(-\ln \mathrm{v}\right)}^{\uptheta}\right]}^{\frac{1}{\uptheta}}}$$ 1 ≤ θ
Joe (Archimedean) $$1-{\left({\left(1-\mathrm{u}\right)}^{\uptheta}+{\left(1-\mathrm{v}\right)}^{\uptheta}-{\left(1-\mathrm{u}\right)}^{\uptheta}{\left(1-\mathrm{v}\right)}^{\uptheta}\right)}^{\frac{1}{\uptheta}}$$ 1 ≤ θ
Normal (Elliptical) $$\underset{-\infty }{\overset{\varnothing^{-1}\left(\mathrm{u}\right)}{\int }}\underset{-\infty }{\overset{\varnothing^{-1}\left(\mathrm{v}\right)}{\int }}\frac{1}{2\uppi {\left(1-{\uprho}^2\right)}^{\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{2}\right.}}{\mathrm{e}}^{-\frac{{\mathrm{u}}^2-2\uprho \mathrm{uv}+{\mathrm{v}}^2}{2\left(1-{\uprho}^2\right)}}\mathrm{dudv}$$ −1 ≤ ρ ≤ 1
Student t (Elliptical) $$\underset{-\infty }{\overset{{{\mathrm{t}}_{\mathrm{df}}}^{-1}\left(\mathrm{u}\right)}{\int }}\underset{-\infty }{\overset{{{\mathrm{t}}_{\mathrm{df}}}^{-1}\left(\mathrm{v}\right)}{\int }}\frac{1}{2\uppi {\left(1-{\uprho}^2\right)}^{\raisebox{1ex}{1}\!\left/ \!\raisebox{-1ex}{2}\right.}}{\left(1+\frac{{\mathrm{u}}^2-2\uprho \mathrm{uv}+{\mathrm{v}}^2}{\mathrm{df}\left(1-{\uprho}^2\right)}\right)}^{-\frac{\mathrm{df}+2}{2}}\mathrm{dudv}$$ −1 ≤ ρ ≤ 1 df ≥ 1