I know that a family of Gaussian copulas generates a standard bivariate normal distribution if and only if the marginal ones are standard normal. This characterizes the Gaussian copulas, where I have a problem is, if I change the standard normals for normals with mean $\mu$ and variance $\sigma^2$, do I get a family of parametric copulas?
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The point of copulas is that they do not care about the margins. Therefore, if you want a Gaussian copula as the dependence structure between the margins and margins that are normal but not standard normal, go for it. Just specify the parameters of the Gaussian copula, and then specify the means and variances of the margins. No additional theory is needed.
To see why this would be a parametric family, the parameters of the Gaussian copula are the off-diagonal elements of the (population-level) correlation matrix, and the parameters of the margins are the usual (population-level) means and variances.
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$\begingroup$ Thank you very much, yes in general I was looking for a "standard" way to arrive at the form of a known parametric family, however, I see that it does not exist $\endgroup$ Mar 14 at 20:08
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$\begingroup$ How do you figure there is no parametric family? A multivariate Gaussian distribution is characterized by its correlations (which are the parameters of the Gaussian copula) and by the means and variances of the margins. $\endgroup$– DaveMar 14 at 20:09
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1$\begingroup$ You could state already in the answer that the parameters of the Gaussian copula will be just the correlation matrix. $\endgroup$ Mar 14 at 20:33