1
$\begingroup$

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

$\endgroup$
1
$\begingroup$

It's the jacobian.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.