VineCopulaNegLL

Computing the negative log-likelihood of a vine copula

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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.