Gumbel

Symmetric to main diagonal

cdf

Following [ACFB]:

$C(u_{1},u_{2})=\exp\left(- \left((-\ln u_{1})^{\theta} + (-\ln u_{2})^{\theta}\right)^{\frac{1}{\theta}}\right),\\$ $\theta\in [1,\infty)$

h-function

$\begin{aligned} h(u_{1},u_{2})&=\frac{\partial C(u_{1},u_{2})}{\partial u_{2}}\\ &=C(u_{1},u_{2})\left( -\frac{1}{\theta} \right)\left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}-1}\theta(-\ln u_{2})^{\theta-1}(-\frac{1}{u_{2}})\\ &=C(u_{1},u_{2})\frac{1}{u_{2}}(-\ln u_{2})^{\theta-1}\left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}-1} \end{aligned}$

pdf

$\begin{aligned} c(u_{1},u_{2})=&\frac{\partial h(u_{1},u_{2})}{\partial u_{1}} \\ =&C(u_{1},u_{2})\frac{1}{u_{1}}(-\ln u_{2})^{\theta-1}\left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta}\right)^{\frac{1}{\theta}-1} \\ &\cdot\frac{1}{u_{2}}(-\ln u_{2})^{\theta-1}\left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}-1}\\ &+C(u_{1},u_{2})\frac{1}{u_{2}}(-\ln u_{2})^{\theta-1}\left( \frac{1}{\theta}-1 \right) \left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}-2}\\ &\cdot \theta(-\ln u_{1})^{\theta-1}\left( -\frac{1}{u_{1}} \right) \\ =&C(u_{1},u_{2})(u_{1}u_{2})^{-1}\left( (-\ln u_{1})(-\ln u_{2}) \right)^{\theta-1}\\ &\cdot\left( \left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}+\frac{1}{\theta}-2} - \left( \frac{1}{\theta}-1 \right)\theta \left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta}\right)^{\frac{1}{\theta}-2}\right) \\ =&C(u_{1},u_{2})(u_{1}u_{2})^{-1}\left( (-\ln u_{1})(-\ln u_{2}) \right)^{\theta-1}\left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta}\right)^{\frac{1}{\theta}-2}\\ &\cdot\left( \left( (-\ln u_{1})^{\theta}+(-\ln u_{2})^{\theta} \right)^{\frac{1}{\theta}} + \theta - 1 \right) \end{aligned}$

v-function

$\begin{aligned} v(u_{2},u_{1})&=h(u_{2}, u_{1}) \end{aligned}$

inverse h-function

Not solveable.

inverse v-function

Not solveable.

$v^{-1}(u_{2},u_{1})=h^{-1}(u_{2},u_{1})$

Copula theory

Notes on copula theory

Gumbel

cdf

h-function

pdf

v-function

inverse h-function

inverse v-function

References