Clayton

Symmetric to main diagonal

cdf

Following [ACFB]:

$C(u_{1},u_{2})=\left( u_{1}^{-\theta} + u_{2}^{-\theta} -1 \right)^{-\frac{1}{\theta}},\\$ $0<\theta$

h-function

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

pdf

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

v-function

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

inverse h-function

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

inverse v-function

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

Copula theory

Notes on copula theory

Clayton

cdf

h-function

pdf

v-function

inverse h-function

inverse v-function

References