Remark
The package VineCopula is a continuation of the package CDVine by U. Schepsmeier and E. C. Brechmann. It includes all functions
implemented in CDVine for the bivariate case (BiCop-functions).Bivariate copula families
In this package several bivariate copula families are included for bivariate analysis as well as for multivariate analysis using vine copulas.
It provides functionality of elliptical (Gaussian and Student-t) as well as Archimedean (Clayton, Gumbel, Frank, Joe, BB1, BB6, BB7 and BB8) copulas
to cover a large bandwidth of possible dependence structures.
For the Archimedean copula families rotated versions are included to cover negative dependence too.
The two parameter BB1, BB6, BB7 and BB8 copulas are however numerically instable for large parameters,
in particular, if BB6, BB7 and BB8 copulas are close to the Joe copula which is a boundary case of these three copula families.
In general, the user should be careful with extreme parameter choices.
The following table shows the parameter ranges of bivariate copula families with parameters par and par2:
lll{
Copula family par par2
Gaussian $(-1,1)$ -
Student t $(-1,1)$ $(2,\infty)$
(Survival) Clayton $(0,\infty)$ -
(Survival) Gumbel $[1,\infty)$ -
Frank $R\backslash{0}$ -
(Survival) Joe $(1,\infty)$ -
Rotated Clayton (90 and 270 degrees) $(-\infty,0)$ -
Rotated Gumbel (90 and 270 degrees) $(-\infty,-1]$ -
Rotated Joe (90 and 270 degrees) $(-\infty,-1)$ -
(Survival) Clayton-Gumbel (BB1) $(0,\infty)$ $[1,\infty)$
(Survival) Joe-Gumbel (BB6) $[1,\infty)$ $[1,\infty)$
(Survival) Joe-Clayton (BB7) $[1,\infty)$ $(0,\infty)$
(Survival) Joe-Frank (BB8) $[1,\infty)$ $(0,1]$
Rotated Clayton-Gumbel (90 and 270 degrees) $(-\infty,0)$ $(-\infty,-1]$
Rotated Joe-Gumbel (90 and 270 degrees) $(-\infty,-1]$ $(-\infty,-1]$
Rotated Joe-Clayton (90 and 270 degrees) $(-\infty,-1]$ $(-\infty,0)$
Rotated Joe-Frank (90 and 270 degrees) $(-\infty,-1]$ $[-1,0)$
}R-vine copula models
The specification of an R-vine is done in matrix notation, introduced by Dissmann et al. (2011). One matrix contains the R-vine tree structure,
one the copula families utilized and two matrices corresponding parameter values.
These four matrices are stored in an RVineMatrix object created by the function RVineMatrix. Each matrix is a d x d lower triangular matrix.
Since C- and D-vines are special cases, boundary cases, of R-vines one can write each C- or D-vine in R-vine notation. The transformation
of notation to or from an R-vine can be done via C2RVine, D2RVine, R2CVine and R2DVine, which provide an interface to the package CDVine.
For more details see the documentation of the functions.Acknowledgment
We acknowledge substantial contributions by our working group at Technische Universitaet Muenchen,
in particular by Carlos Almeida and Aleksey Min.
In addition, we like to thank Shing (Eric) Fu, Feng Zhu, Guang (Jack) Yang, and Harry Joe for providing their implementation
of the method by Knight (1966) for efficiently computing the empirical Kendall's tau.
We are especially grateful to Harry Joe for his contributions to the implementation of the bivariate Archimedean copulas.