Table 1 Copula families of distribution functions
From: Multivariate distributions of correlated binary variables generated by pair-copulas
Name | Copula function | Parameter range |
|---|---|---|
Gaussian | C1(u1,u2;γ)=Φ2(Φ−1(u1),Φ−1(u2);γ) | − 1≤γ≤1 |
Clayton | \(C_{2}(u_{1},u_{2};\alpha)=\max ((\,u_{1}^{-\alpha }+u_{2}^{-\alpha }-1)\,^{-\frac {1}{\alpha }},0)\) | α∈[ −1,∞)∖{0} |
Frank | \(C_{3}(u_{1},u_{2};\alpha)=-\frac {1}{\alpha }\log (1+\frac {(e^{-\alpha u_{1}}-1)(e^{-\alpha u_{2}}-1)}{e^{-\alpha }-1})\) | α≠0 |
Gumbel | \(C_{4}(u_{1},u_{2};\alpha)=e^{-[\,(-\log u_{1})^{\alpha }+(-\log u_{2})^{\alpha }]\,^{\frac {1}{\alpha }}}\) | α≥1 |
Independent | C5(u1,u2)=u1∗u2 | N/A |