Table 3 Copula estimation of parameters and structure dependence

Gaussian

\({C}_{N}\left(u,v,\rho \right)=\varnothing ({\varnothing }^{-1}\left(u\right), {\varnothing }^{-1}\left(v\right))\)

\(\rho\)

No tail dependence:\({\lambda }_{U}={\lambda }_{L}=0\)

Clayton

\({C}_{C}\left(u,v,\theta \right)={C}_{C}(1-u, 1-v;\theta )\)

\(\theta\)

Asymmetric tail dependence: \({\lambda }_{U}=0\),\({\lambda }_{L}={2}^{-1/\theta }\)

Gumbel

\({C}_{G}\left(u,v,\delta \right)={\text{exp}}(-{\left({\left(-{\text{log}}(u)\right)}^{\delta }+{\left(-{\text{log}}(v)\right)}^{\delta }\right)}^{1/\delta }\))

\(\delta \ge 1\)

Asymmetric tail dependence: \({\lambda }_{U}=2-{2}^{1/\delta }\),\({\lambda }_{L}=0\)