Gaussian copula

Parameter restrictions: $-1<\rho<1$

cdf

Following [Joe14]:

$C(u_{1},u_{2})=\Phi_{2}\left( \Phi^{-1}(u_{1}), \Phi^{-1}(u_{2})\right),$

where $\Phi_{2}$ denotes the cdf of the 2-dimensional normal distribution with correlation coefficient $\rho$, and $\Phi$ denotes the cdf of the univariate standard normal distribution.

pdf

$c(u_{1},u_{2})=\frac{\phi_{2}(\Phi^{-1}(u_{1}), \Phi^{-1}(u_{2}))}{\phi(\Phi^{-1}(u_{1}))\phi(\Phi^{-1}(u_{2}))},$

where $\phi_{2}$ denotes the pdf of the 2-dimensional normal distribution with correlation coefficient $\rho$, and $\phi$ denotes the pdf of the univariate standard normal distribution.

Copula theory

Notes on copula theory

Gaussian copula

cdf

pdf

References