From first reading these vines piggyback off the bivariate copulas to get multivariate ones without the complication of actually trying to derive those. This is similar to how bivariate copulas generalized bivariate distributions by using univariate marginals.
I’ve written up the multivariate normal copula and other bivariate ones. I was really more curious with a simple example of how one would code up, say, connecting two different bivariate copula families or a bivariate copula and a univariate distribution. Simple examples that build up intuition to code up the more general cases.
@moderators I propose moving the last few replies to a new topic. I don’t know how to bulk move. A separate topic starting with @jmh530’s post on vines would be better. And linking back here.
I agree that simple examples help build intuition. That’s why I would do it first using the bivariate normal copula likelihoods and check that the vine version produces the same results. Then you can write a non-normal bivariate copula likelihood and see what happens when you change one of them.
Check out pages 9 and 10 of this
I think that makes it pretty clear on how the decomposition is happening. Page 9 would require you to condition variables 1 and 3 on variable 2, choose a density for that relationship, and then fit a copula. C/R/D vine is just about how to choose the decomposition.