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 fastly 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 copulas and a lower number of conditional pair-copulas. In our earlier papers we introduced the concept of cherry-tree copulas. In this paper we characterize the relation between cherry tree copulas and truncated R-vine copulas. It turns out that the concept of cherry-tree copula is more general than the concept of truncated R-vine copula. Although both contain in their expressions conditional independences between the variables, the truncated R-vines constructed in greedy way do not exploit the existing conditional independences in the data. We give a necessary and sufficient condition for a cherry-tree copula to be a truncated R-vine copula. We introduce a new method for truncated R-vine modeling. The new idea is that in the first step we construct the top tree by exploiting conditional independences for finding a good-fitting cherry-tree of order k. If this top tree is a tree in an R-vine structure then this will define a truncated R-vine at level k and in the second step we construct a sequence of trees which leads to it. If this top tree is not a tree in an R-vine structure then we can transform it into such a tree at level k + 1 and then we can again apply the second step. The second step is performed by a backward construction named Backward Algorithm. This way the cherry-tree copulas always can be expressed by pair-copulas and conditional pair-copulas.