On the connection between cherry-tree copulas and truncated R-vine copulas

Vine copulas are a flexible way for modeling dependences using only pair-copulas as building blocks. However if the number of variables grows the problem gets fast intractable. For dealing with this problem Brechmann at al. proposed the truncated R-vine copulas. The truncated R-vine copula has the very useful property that it can be constructed by using only pair-wise copulas, and conditional pair-wise copulas. In our earlier papers we introduced the concept of cherry-tree copulas. In this paper we characterize the relation between the cherry-tree copulas and the truncated R-vine copulas. Both are based on exploiting of some conditional independences between the variables. We give a necessary and sufficient condition for a cherry-tree copula to be a truncated R-vine copula. We introduce a new perspective for truncated R-vine modeling. The new idea is finding first a good fitting cherry-tree copula of order $k$. Then, if this is also a truncated R-vine copula we apply the Backward Algorithm introduced in this paper. This way the construction of a sequence of trees which leads to it becomes possible. So the cherry-tree copula can be expressed by pair-copulas and conditional pair-copulas. In the case when the fitted $k$ order cherry-tree copula is not a truncated R-vine copula we give an algorithm to transform it into truncated R-vine copula at level $k+1$. Therefore this cherry-tree copula can also be expressed by pair-copulas.