Example

As pair copula constructions become quite complicated in higher dimensions, we want to introduce most of the theory with the aid of a concrete example. In this example, we look at the joint distribution of 7 variables. The following network visualization shall give a first impression of how individual variables get connected in our pair copula construction. Nevertheless, this graph is in no way unique, so that any visual characteristics like proximity could be due to graphical artifacts only. Also, we only show conditional links up to layer 3:

rvine matrix conversion

R-Vines

Transforming an R-Vine matrix to tree notation:

rvine matrix conversion

Standard notation

We will use the example of Dißmann (page 31):

rvine trees of different layers

As Dißmann13 shows, this vine decomposition can be expressed as R-Vine matrix. Two possible R-Vine matrices are (page 36):

$\begin{pmatrix} 4 & & & & & & \\ 7 & 5 & & & & & \\ 6 & 7 & 1 & & & & \\ 5 & 6 & 7 & 7 & & & \\ 1 & 1 & 6 & 2 & 6 & & \\ 2 & 3 & 3 & 3 & 2 & 2 & \\ 3 & 2 & 2 & 6 & 3 & 3 & 3 \end{pmatrix}$ $\begin{pmatrix} 7 & & & & & & \\ 4 & 4 & & & & & \\ 5 & 6 & 6 & & & & \\ 1 & 5 & 5 & 5 & & & \\ 2 & 1 & 1 & 1 & 1 & & \\ 3 & 2 & 2 & 3 & 3 & 3 & \\ 6 & 3 & 3 & 2 & 2 & 2 & 2 \end{pmatrix}$

Alternative notation

$\begin{pmatrix} 0 & 2 & 2 & 2 & 3 & 2 & 2 \\ 1 & 0 & 3 & 3 & 5 & 3 & 3 \\ 2 & 2 & 0 & 4 & 2 & 6 & 6 \\ 3 & 3 & 3 & 0 & 1 & 5 & 5 \\ 3 & 2 & 2 & 1 & 0 & 1 & 1 \\ 3 & 3 & 3 & 5 & 1 & 0 & 7 \\ 6 & 6 & 6 & 6 & 6 & 6 & 0 \end{pmatrix}$

Copula theory

Notes on copula theory

Example

R-Vines

Standard notation

Alternative notation

References