How many pairwise copulas in a $k$-dimensional R-vine?

Question

I am studying R-vine copulas. Let $k$ denote the number of random variables the joint distribution of which we are modeling. R-vine breaks down the $k$-variate copula into $K$ bivariate copulas. I am trying to understand how $K$ is related to $k$.

Question: How many pairwise copulas are there in a $k$-dimensional R-vine?

score 1 · Answer 1 · answered Apr 19 '21 at 09:30

1

My guess is, $K=k(k-1)/2$.

The tree structure in Figure 1 of Dissmann et al. (2013) illustrates that an R-vine is represented by $k-1$ trees that have $k-1,k-2,\dots,1$ edges, respectively, where each edge stands for a pair copula. This is an arithmetic progression the sum of which is: $(k-1)+(k-2)+\dots+1=k(k-1)/2$.
(My uncertainty is, is this the general case, or is it just a special case. It seems to me it is the general case.)

Reference

Dissmann et al. "Selecting and estimating regular vine copulae and application to financial returns" (2013) (arXiv version here).

answered Apr 19 '21 at 09:30

Richard Hardy

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Yes, it is correct. – Maryam May 19 '21 at 14:19
Kindly note that `k(k-1)/2` is the rule for all vine copula models. However, and especially for high dimensional data, you may need to reduce the complexity of the model by truncating the vine model. – Maryam May 19 '21 at 18:14
@Maryam, thank you! – Richard Hardy May 19 '21 at 18:39

How many pairwise copulas in a $k$-dimensional R-vine?

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