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I was wondering if the graphical methods can be used instead of formal tests such as Cramer-Von Mises test as the GOF for copulas?

The scatter plot of pseudo-observations is as follows:

enter image description here

The Q-Q plot of non-parametric Kendall's tau distribution versus the parametric distribution of fitted copulas (Gumbel, Clayton, Frank, and Joe) is shown as follows.

enter image description here

It seems that the Joe copula is suitable for the dataset. Also the contour plot of fitted Joe copula overlaid the contour plot of the empirical copula shows the suitability of this copula in defining the bivariate distribution of the dataset compared with the other copulas.

enter image description here

As Joe copula is more suitable in expressing the empirical copula, I also compared the scatter plot of pseudo-observations with the generated random numbers of Joe copula: enter image description here

Now, based on these graphical methods can we choose the Joe copula as the suitable form of copula for my dataset?

I appreciate any advise.

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Yes, we can use scatter plot as a method to choose the bivariate copula. However, if your data is mixture (more than one copula family) hence, this method can be very difficult in practice. Then, what you did in comparing the simulated with the original one is very good method to double check your result.

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  • $\begingroup$ Thank you for the response. So, how can I know if my data is a mixture of more than one copula family? $\endgroup$
    – Saba Ghotbi
    Commented Jan 6, 2020 at 15:05
  • $\begingroup$ You can easily know that from the scatter plot. For ex, if you have two different strong tails in two different corners! And that is not a t-copula. $\endgroup$
    – Maryam
    Commented Jan 9, 2020 at 9:43
  • $\begingroup$ @Maryam the op overlaid the contour plot of a fitted Joe copula over the contour plot of the empirical copula to show its suitability in defining the bivariate distribution of the dataset. Why bother simulating a copula when he can just use the actual empirical copula? $\endgroup$
    – develarist
    Commented Sep 6, 2020 at 13:20

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