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I'm really struggling with bivariate copula's. Long story short, I can only use Gaussian copulas. I'm therefore interested in the joint PDF for which the Gaussian copula can be applied.

So for example:

  • The Gumbel copula is used for extreme distributions.
  • The Gaussian copula is used for linear correlation.
  • The Archimedean copula and the t-copula are used for dependence in the tail.

So this clearly "limits" the joint PDFs for which the Gaussian Copula can be used to accurately model, and there are many cases that it simply won't apply. I.e., the joint PDF must have linear correlation, small or no tail dependence, and it is not ideal for the extreme distributions. Are there any more restrictions?

Furthermore, does anyone know of a simple joint PDF that can be accurately used? I'd ideally like to find one which involves Uniform, Gamma and Beta distributions. Several sources I've looked at just plough through this and state that the Gaussian copula can be used all over the place–but I just can't see how?! Have these authors just gotten carried away and ignored the limitations?

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It sounds like you are interested in fitting bivariate distributions to data? One way of doing this is to fit a bivariate normal distribution. Unfortunately, many bivariate data sets do not look like a bivariate normal at all. So people have considered more general distributions. One approach is to consider the marginal distributions separately, and then describe the dependence structure using a copula.

Given a choice of marginals, you then have to think about which copula might be appropriate. Unfortunately, we are still living in the copula stone age. Our choice of copulas basically boils down to a list of possibilities which have been written down by people. We choose one of these, and see how well it fits our data, although there is no universally-accepted way of doing so except for grunting and pointing. If it doesn't fit very well, then we can try another, and keep going until we are happy.

Usually, instead of choosing one particular copula, you choose a family of copulas depending on one or more parameters, and then try to find the copula in this family which gives the best fit. One such family is the family of Gaussian copulas, which depend on a parameter $\rho$. By using this family, you are making the assumption that, if $F$ and $G$ are your marginal cdfs, then $\Phi^{-1}F(X)$ and $\Phi^{-1}G(Y)$ follow a bivariate normal distribution with correlation $\rho$. This is perhaps what you mean by linear correlation?

Essentially, this just means that you are transforming $X$ and $Y$ separately (by choosing the marginals) and then fitting a bivariate normal to the result.

If you had the data in @Glen_b's example then you would observe that the $Y$ variable, $u$, looks like it has a uniform distrbution. So you would transform it using $\Phi^{-1}$, the inverse of the normal cdf. On the other hand, the $X$ variable, $x$, looks normal, so you would not transform it. Then you would try to fit a bivariate normal to the transformed data.

enter image description here

It turns out that, for real-life data, this is often inadequate; there is no pair of univariate transformations that will make $X$ and $Y$ look like a bivariate normal distribution and so the Gaussian copula is not a good choice. In particular, this happens when you have tail dependence. Here is a blog post on the Gaussian copula by TGR which goes into more details.

Using the Gaussian copula might work better than just using a bivariate normal, but there might also be better choices available, particularly if you care about the tails of your distribution. For example, there are many popular families of archimedean copulas which have tail dependence.

On the other hand, the Gaussian copula could be a perfectly good choice of copula if you want something flexible and easy to fit and don't mind about the tails. It depends on what it's being used for.

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  • $\begingroup$ @Glen_b Thanks for the bounty! It was most generous of you. $\endgroup$ – Flounderer Jun 30 '16 at 7:07
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What convinces you that the Gaussian copula only applies to linear correlation?

This would seem to be a counterexample: a pair of variables with a Gaussian copula, but they're not linearly related:

enter image description here

If that's not what you mean by 'linear correlation', you will need to be more explicit

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