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When I was looking at the Gaussian Copula Example @ http://en.wikipedia.org/wiki/Copula_(probability_theory)

I realized the Gaussian Copula is stated as follow \begin{equation} C^{Gauss}_\Sigma (u) = \frac{1}{\sqrt\det{\Sigma}} \exp{\Bigg ( -\frac{1}{2} \begin{pmatrix} \Phi^{-1}(u_1) \\ \dots \\ \Phi^{-1}(u_d)\end{pmatrix}^T. (\Sigma^{-1} - I).\begin{pmatrix} \Phi^{-1}(u_1) \\ \dots \\ \Phi^{-1}(u_d)\end{pmatrix} \Bigg) } \end{equation} where $\Sigma$ is the correlation matrix, $\Phi^{-1}$ is the inverse cumulative distribution function of a standard normal and $I$ is the identity matrix.

The question is, why is there an identity matrix in the exponential form?

Thank you

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2 Answers

It's the jacobian.

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It is the Jacobian which appears in the density. Really there should be some $\text{d}u$ terms to make it clearer that it is not the probability distribution function.

What's a good reference for this derivation (online)? Wikipedia only has an advertisement for someone's book as a reference.

I find this presentation very confusing, for example I had been missing that this was the density (small c) as opposed to the CDF and was obviously not making any sense of it. I can imagine other readers doing the same thing.

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