BiCopPar2TailDep: Tail dependence coefficients of a bivariate copula
In CDVine: Statistical Inference of C- And D-Vine Copulas

Description Usage Arguments Value Author(s) References See Also Examples

This function computes the theoretical tail dependence coefficients of a bivariate copula for given parameter values.

1	BiCopPar2TailDep(family, par, par2=0)

`family`	An integer defining the bivariate copula family: `0` = independence copula `1` = Gaussian copula `2` = Student t copula (t-copula) `3` = Clayton copula `4` = Gumbel copula `5` = Frank copula `6` = Joe copula `7` = BB1 copula `8` = BB6 copula `9` = BB7 copula `10` = BB8 copula `13` = rotated Clayton copula (180 degrees; “survival Clayton”) `14` = rotated Gumbel copula (180 degrees; “survival Gumbel”) `16` = rotated Joe copula (180 degrees; “survival Joe”) `17` = rotated BB1 copula (180 degrees; “survival BB1”) `18` = rotated BB6 copula (180 degrees; “survival BB6”) `19` = rotated BB7 copula (180 degrees; “survival BB7”) `20` = rotated BB8 copula (180 degrees; “survival BB8”) `23` = rotated Clayton copula (90 degrees) `24` = rotated Gumbel copula (90 degrees) `26` = rotated Joe copula (90 degrees) `27` = rotated BB1 copula (90 degrees) `28` = rotated BB6 copula (90 degrees) `29` = rotated BB7 copula (90 degrees) `30` = rotated BB8 copula (90 degrees) `33` = rotated Clayton copula (270 degrees) `34` = rotated Gumbel copula (270 degrees) `36` = rotated Joe copula (270 degrees) `37` = rotated BB1 copula (270 degrees) `38` = rotated BB6 copula (270 degrees) `39` = rotated BB7 copula (270 degrees) `40` = rotated BB8 copula (270 degrees)
`par`	Copula parameter.
`par2`	Second parameter for the two parameter t-, BB1, BB6, BB7 and BB8 copulas (default: `par2 = 0`).

lower

Lower tail dependence coefficient of the given bivariate copula family C:

λ_L = lim_{u->0} C(u,u)/u

upper

Upper tail dependence coefficient of the given bivariate copula family C:

λ_U = lim_{u->1}(1-2u+C(u,u))/(1-u)

Lower and upper tail dependence coefficients for bivariate copula families and parameters (θ for one parameter families and the first parameter of the t-copula with ν degrees of freedom, θ and δ for the two parameter BB1, BB6, BB7 and BB8 copulas) are given in the following table.

No.	Lower tail dependence	Upper tail dependence
`1`	-	-
`2`	2t_{ν+1}(-√{ν+1}√{(1-θ)/(1+θ)})	2t_{ν+1}(-√{ν+1}√{(1-θ)/(1+θ)})
`3`	2^{-1/θ}	-
`4`	-	2-2^{1/θ}
`5`	-	-
`6`	-	2-2^{1/θ}
`7`	2^{-1/(θδ)}	2-2^{1/δ}
`8`	-	2-2^{1/(θδ)}
`9`	2^{-1/δ}	2-2^{1/θ}
`10`	-	2-2^{1/θ} if δ=1 otherwise 0
`13`	-	2^{-1/θ}
`14`	2-2^{1/θ}	-
`16`	2-2^{1/θ}	-
`17`	2-2^{1/δ}	2^{-1/(θδ)}
`18`	2-2^{1/(θδ)}	-
`19`	2-2^{1/θ}	2^{-1/δ}
`20`	2-2^{1/θ} if δ=1 otherwise 0	-
`23, 33`	-	-
`24, 34`	-	-
`26, 36`	-	-
`27, 37`	-	-
`28, 38`	-	-
`29, 39`	-	-
`30, 40`	-	-

Eike Brechmann

Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.

BiCopPar2Tau

## Example 1: Gaussian copula
BiCopPar2TailDep(1,0.7)

## Example 2: t copula
BiCopPar2TailDep(2,0.7,4)

The CDVine package is no longer developed actively.
Please consider using the more general VineCopula package
(see https://CRAN.R-project.org/package=VineCopula),
which extends and improves the functionality of CDVine.

$lower
[1] 0

$upper
[1] 0

$lower
[1] 0.390684

$upper
[1] 0.390684