Table 2 Basic properties of applied copulas
From: Bivariate hydrologic risk analysis for the Xiangxi River in Three Gorges Reservoir Area, China
Copula Name | Function[\(C_{\theta } (u_{1} ,\;u_{2} )\)] | \(\theta \in\) | Generating functions [\(\phi (t)\)] | \(\tau = 1 + 4\int_{0}^{1} {\frac{\phi (t)}{{\phi ^{\prime}(t)}}dt}\) |
|---|---|---|---|---|
Ali-Mikhail-Haq | \(\frac{{u_{1} u_{2} }}{{[1 - \theta (1 - u_{1} )(1 - u_{2} )]}}\) | [− 1, 1) | \(\ln (\frac{[1 - \theta (1 - t)]}{t})\) | \(\frac{3\theta - 2}{\theta } - [\frac{2}{3}(1 - \theta^{ - 1} )^{2} \ln (1 - \theta )]\) |
Cook-Johnson | \([u_{1}^{ - \theta } + u_{2}^{ - \theta } - 1]^{ - 1/\theta }\) | [− 1, ∞)\{0} | \(t^{ - \theta } - 1\) | \(\frac{\theta }{\theta + 2}\) |
Gumbel-Hougaard | \(\exp \{ - [( - \ln u_{1} )^{\theta } + ( - \ln u_{2} )^{\theta } ]^{1/\theta } \}\) | [1, ∞) | \(( - \ln t)^{\theta }\) | \(1 - \theta^{ - 1}\) |
Frank | \(- \frac{1}{\theta }\ln \{ 1 + \frac{{(e^{ - \theta u} - 1)(e^{ - \theta v} - 1)}}{{e^{ - \theta } - 1}}\}\) | [− ∞, ∞)\{0} | \(\ln [\frac{{e^{ - \theta t} - 1}}{{e^{ - \theta } - 1}}]\) | \(1 - \frac{4}{\theta }[D_{1} ( - \theta ) - 1]\) a |