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I have a bivariate sample in the [0,1] square for which I am trying to find the copula that best describes it. (I am new to copulas.)

So far, I have tried all classes in the "copula" R package. Using ML, the best fit was the t-Copula. However, this clearly does not do a great job in describing my data; see the image below that plots the raw data (top rows) and t-Copula fit (bottom rows).

Bivariate data and copula fit (Student t)

I have also tried the 'opt_auto' function from the "cylcop" package, which suggested a von Mises copula that, again, doesn't represent my data well.

The main problem is that the fits do not capture the asymmetric density on the main diagonal (significantly higher concentration of probabilities around (0,0) compared to around (1,1)).

Does anyone have an idea regarding what to try next?

kjetil b halvorsen
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dganghel
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  • @Moya True, I guess I should move the question to stats, thanks! Regarding the Frank copula though, I already tried it (it is part of the copula package) and it performs worse than the Student t. – dganghel Jan 05 '24 at 16:04
  • Look at https://stats.stackexchange.com/questions/88661/what-is-copula-transformation, https://stats.stackexchange.com/questions/557556/how-are-copulas-used-in-the-real-world, – kjetil b halvorsen Jan 06 '24 at 21:39
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    It seems that a single available copula may not fit your data. However, you may try to fit a mixture of Joe and Frank copula. It will not show a great fit, but it is worth trying. – Maryam Jan 07 '24 at 13:20
  • What is your ultimate goal, and how is this goal related to the name of a copula? Why do you think one of the few named copulas will fit your data, and what do you mean by "best described"? – g g Jan 09 '24 at 08:17
  • @kjetilbhalvorsen Those posts do not help with answering my question. – dganghel Jan 09 '24 at 14:51
  • @Maryam Thanks for the suggestion! I have tried some mixtures but with no luck thus far; I will continue to look into them and return with an update. I have also found this paper https://doi.org/10.1016/j.jmva.2018.11.012, which seems very promising. – dganghel Jan 09 '24 at 14:51
  • @gg: I don't care about the name of the copula, I only care about finding one that best explains my data, by that meaning having the best goodness of fit measure(s) compared to a set of candidates. In particular, as the figure suggests, I am looking for a good way to model the heavy left tail dependence between my two variables. I certainly do not think that "one of the few named copulas will fit your data", I am unsure why you say that. At this point, I think complicating the question by providing numerical results doesn't help in any way, the figure should convey well what am I looking for. – dganghel Jan 09 '24 at 14:58

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