# VineCopulaNegLL

Computing the negative log-likelihood of a vine copula

# Purpose

The function computes the value of the log-likelihood for a given matrix of observations u, which have to lie in the d-dimensional unit cube. Possible Vine copula types:

```
0 C-Vine
1 D-Vine
```

# Usage

```
Simplified standard C-Vine or D-Vine copula
CLL = CopulaNegLL('C-Vine',u,families,thetas,d)
CLL = CopulaNegLL('D-Vine',u,families,thetas,d)
Simplified standard C-Vine or D-Vine copula with rotated copulas
CLL = CopulaNegLL('C-Vine',u,families,thetas,d,rotation)
CLL = CopulaNegLL('D-Vine',u,families,thetas,d,rotation)
Truncated simplified standard C-Vine or D-Vine copula
CLL = CopulaNegLL('C-Vine',u,families,thetas,d,rotation,CutOffTree)
CLL = CopulaNegLL('D-Vine',u,families,thetas,d,rotation,CutOffTree)
```

# Inputs

```
type = The vine copula type.
u = A (n x d) dimensional vector of values lying in [0,1]
(the observations).
families = A vector of the pair-copula families, which
are part of the PCC. The vector has to have
the length (d-1)*d/2. The first d-1 entries are
the copula families in the first tree and the
next d-2 entries are the copula families in the
second tree and so on. That means, for d=4 the
array should look similar to this {'Frank',
'Frank', 'Frank', 'AMH', 'AMH', 'Clayton'}, which
is the special case where all copulas in the
first tree are Frank copulas, all copulas in the
second tree are AMH copulas and all copulas in
the third tree are Clayton copulas.
The order of the families is:
* Exemplarily for the four-dimensional C-Vine):
C12, C13, C14, C23|1, C24|1, C34|12
* Exemplarily for the four-dimensional D-Vine):
C12, C23, C34, C13|2, C24|3, C14|23
Note: If families is a simple string/character,
e.g., 'Clayton', then all pair-copulas are
specified to be from this copula family.
thetas = The values of the parameters for the (d-1)*d/2 pair-
copulas. These parameter values have to be given in
the same order as the families vector, but in a
row-vector. If a pair-copula is a independence
copula, then there is no parameter needed.
Furthermore, if a pair-copula has two or more
parameters, the parameters have to be given in same
order as they have to be provided if the pair-copula
is considered only. For example for a t-copula, the
first parameter is rho and the second parameter is
the degrees of freedom parameter nu.
Note: If thetas is a skalar then all parameters
(i.e., the parameters of all pair-copulas) are set to
the same value (i.e. to the one given).
d = The dimension of the C- or D-Vine.
rotation = A vector of the same dimension as families in
which one can specify rotation levels.
CutOffTree = The CutOffTree (or also called truncation level) can
be used to set all pair-copulas from the (CutOffTree
+ 1)-th tree on to independence copulas (i.e., ignore
them in the joint estimation).
```

# Outputs

```
CLL = The value of the negative log-likelihood for
the data matrix u.
```