As we all know, there are several copula functions, each with its own ability to describe specific dependency structure. I wonder what the Frank copula can tell me. For example, Clayton copula is a lower tail dependency function; that is, the lower values are correlated more strongly than the larger values. However, I could not understand or interpret the plot of Frank. What I will learn from it?
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6$\begingroup$ I would not assume that copulas are something that "we all know," though certainly something we can all learn from! :) $\endgroup$– doubledCommented Jun 4, 2020 at 16:21
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More generally, how do we choose which copula model to use in a given problem? The main guiding principle I learned is to choose a copula model based on the dependence structure of the variables.
Specifically regarding the Frank copula, I'll first introduce two concepts you may know (at a relatively intuitive level, I don't want to start using lots of mathematical notation). A copula is called comprehensive if the copula allows for any dependence structure from full negative rank correlation to full positive rank correlation and also allows for independence. A copula is called archimedean is you can basically model all the dependency of the variables through a generator function.
It can be shown that the only two comprehensive achimedean copulas are Clayton and Frank, so if you want those two properties, you have two choices (I think Nelsen's An Introduction to Copulas has this result). Now how do they differ? Well Clayton is asymmetric, and has more dependence on the negative tail, so you use it when the variables are likely to be jointly low values, but not high. In contrast, Frank is symmetric in the dependence structure, so you use it when the variables are equally likely to be jointly low or high values.
Frank's copula was introduced by Genest (1987), and if you're interested in more motivation for it, I definitely suggest starting by reading that paper:
Genest, Christian. "Frank's family of bivariate distributions." Biometrika 74.3 (1987): 549-555.
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3$\begingroup$ As the parameters of Frank increases, the corolation in the middle increased as well. So, it models the middle values more than the tails. $\endgroup$– AliceCommented Jun 4, 2020 at 17:33
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$\begingroup$ @Glen_b-ReinstateMonica yeah I should have done a better job there. I was trying to make the point that if you knew that each variable is marginally gaussian, the straightforward choice to model joint is to use multivariate gaussian to model, but definitely agree it's not helpful in this explanation, as you could do that for any (cts) random variable. I edited accordingly--Thx! $\endgroup$– doubledCommented Jun 5, 2020 at 13:42