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I have a question regarding Gaussian copulas:

The multivariate Gaussian copula is defined as, $$ C(u_1,\dots,u_n;\Sigma) = \Phi_{\Sigma}(\Phi^{-1}(u_1),\dots,\Phi^{-1}(u_n)), $$ where $\Phi_{\Sigma}$ is a multivariate $n$-dimensional normal distribution with correlation matrix $\Sigma$ and $\Phi$ is the standard univariate cumulative distribution function. How can we show that the corresponding density is: \begin{align} c(u_1,\dots,u_n;\Sigma) &= \frac{1}{\sqrt{\mbox{det} \Sigma}}\exp \begin{pmatrix} - \displaystyle{\frac{1}{2}} \begin{bmatrix} \Phi^{-1}(u_1) \\ \vdots \\ \Phi^{-1}(u_n) \end{bmatrix}^{T} [\Sigma^{-1} - I ] \begin{bmatrix} \Phi^{-1}(u_1) \\ \vdots \\ \Phi^{-1}(u_n) \end{bmatrix} \end{pmatrix}, \end{align} where $I$ is the identity matrix?

Wawel100
  • 153

1 Answers1

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Let $x=[\Phi^{-1}(u_1),\ldots,\Phi^{-1}(u_n)]^{\top}$. Then \begin{align} c(u_1,\ldots,u_n;\Sigma)&=\frac{\partial^nC(u_1,.\ldots,u_n;\Sigma)}{\partial u_1\cdots\partial u_n}=\frac{\Phi(x;0,\Sigma)}{\prod_{i=1}^n \phi(x_i)} \\ &=(2\pi)^{-\frac{n}{2}}|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}\Sigma^{-1}x\right)\times\prod_{i=1}^n (2\pi)^{-\frac{1}{2}}\exp\!\left(\frac{1}{2}x_i^2\right) \\ &=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}\Sigma^{-1}x+\frac{1}{2}x^{\top}x\right) \\ &=|\Sigma|^{-\frac{1}{2}}\exp\!\left(-\frac{1}{2}x^{\top}(\Sigma^{-1}-I)x\right). \end{align}