Asymmetric FGM copula

cdf

Following [Nel06], with $b=0$ :

$C(u_{1},u_{2})=u_{1}u_{2} + \theta u_{1}u_{2}(1 - u_{1} - u_{2} + u_{1}u_{2})u_{2}(1 - u_{1})$

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$c(u_{1},u_{2})= u_{1} \left( \theta u_{1} (6-9 u_{2})u_{2} + \theta(12 u_{2}-8)u_{2} \right) + \theta(2-3 u_{2})u_{2} +1$

hfun

[Wolfram|Alpha]

$h(u_{1},u_{2})=-3\theta u_{1}^{3}u_{2}^{2}+ 2\theta u_{1}^{3}u_{2} + 6\theta u_{1}^{2}u_{2}^{2}-4\theta u_{1}^{2}u_{2} - 3\theta u_{1}u_{2}^{2} + 2\theta u_{1}u_{2} + u_{1}$

vfun

[Wolfram|Alpha]

$v(u_{2},u_{1})= -3\theta u_{1}^{2}u_{2}^{3} + 3\theta u_{1}^{2}u_{2}^{2} + 4\theta u_{1}u_{2}^{3} - 4 \theta u_{1}u_{2}^{2} - \theta u_{2}^{3} + \theta u_{2}^{2} + u_{2}$

Copula theory

Notes on copula theory

Asymmetric FGM copula

cdf

pdf

hfun

vfun

References