Tail Dependence Coefficients of a Bivariate Copula

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

BiCopPar2TailDep(family, par, par2 = 0, obj = NULL, check.pars = TRUE)

Arguments

family: integer; single number or vector of size n; defines 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'') \cr `14` = rotated Gumbel copula (180 degrees; survival Gumbel'')
16 = rotated Joe copula (180 degrees; survival Joe'') \cr `17` = rotated BB1 copula (180 degrees; survival BB1'')
18 = rotated BB6 copula (180 degrees; survival BB6'')\cr `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)
`104` = Tawn type 1 copula
`114` = rotated Tawn type 1 copula (180 degrees)
`124` = rotated Tawn type 1 copula (90 degrees)
`134` = rotated Tawn type 1 copula (270 degrees)
`204` = Tawn type 2 copula
`214` = rotated Tawn type 2 copula (180 degrees)
`224` = rotated Tawn type 2 copula (90 degrees)
`234` = rotated Tawn type 2 copula (270 degrees)
par: numeric; single number or vector of size n; copula parameter.
par2: numeric; single number or vector of size n; second parameter for bivariate copulas with two parameters (t, BB1, BB6, BB7, BB8, Tawn type 1 and type 2; default: par2 = 0). par2 should be an positive integer for the Students's t copula family = 2.
obj: BiCop object containing the family and parameter specification.
check.pars: logical; default is TRUE; if FALSE, checks for family/parameter-consistency are omitted (should only be used with care).

Value

lower: Lower tail dependence coefficient for the given bivariate copula family and parameter(s) par, par2: $$ \lambda_L = \lim_{u\searrow 0}\frac{C(u,u)}{u} $$
upper: Upper tail dependence coefficient for the given bivariate copula family family and parameter(s) par, par2: $$ \lambda_U = \lim_{u\nearrow 1}\frac{1-2u+C(u,u)}{1-u} $$

Lower and upper tail dependence coefficients for bivariate copula families and parameters ($\theta$ for one parameter families and the first parameter of the t-copula with $\nu$ degrees of freedom, $\theta$ and $\delta$ 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`	$2t_{\nu+1}\left(-\sqrt{\nu+1}\sqrt{\frac{1-\theta}{1+\theta}}\right)$	$2t_{\nu+1}\left(-\sqrt{\nu+1}\sqrt{\frac{1-\theta}{1+\theta}}\right)$
`3`	$2^{-1/\theta}$	-
`4`	-	$2-2^{1/\theta}$
`5`	-	-
`6`	-	$2-2^{1/\theta}$
`7`	$2^{-1/(\theta\delta)}$	$2-2^{1/\delta}$
`8`	-	$2-2^{1/(\theta\delta)}$
`9`	$2^{-1/\delta}$	$2-2^{1/\theta}$
`10`	-	$2-2^{1/\theta}$ if $\delta=1$ otherwise 0
`13`	-	$2^{-1/\theta}$
`14`	$2-2^{1/\theta}$	-
`16`	$2-2^{1/\theta}$	-
`17`	$2-2^{1/\delta}$	$2^{-1/(\theta\delta)}$
`18`	$2-2^{1/(\theta\delta)}$	-
`19`	$2-2^{1/\theta}$	$2^{-1/\delta}$
`20`	$2-2^{1/\theta}$ if $\delta=1$ otherwise 0	-
`23, 33`	-	-
`24, 34`	-	-
`26, 36`	-	-
`27, 37`	-	-
`28, 38`	-	-
`29, 39`	-	-
`30, 40`	-	-
`104,204`	-	$\delta+1-(\delta^{\theta}+1)^{1/\theta}$
`114, 214`	$1+\delta-(\delta^{\theta}+1)^{1/\theta}$	-
`124, 224`	-	-
`134, 234`	-	-

Details

If the family and parameter specification is stored in a BiCop object obj, the alternative version

BiCopPar2TailDep(obj)

can be used.

Note

The number n can be chosen arbitrarily, but must agree across arguments.

References

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

Author

Eike Brechmann

Examples

## Example 1: Gaussian copula
BiCopPar2TailDep(1, 0.7)
#> $lower
#> [1] 0
#> 
#> $upper
#> [1] 0
#> 
BiCop(1, 0.7)$taildep  # alternative
#> $lower
#> [1] 0
#> 
#> $upper
#> [1] 0
#> 

## Example 2: Student-t copula
BiCopPar2TailDep(2, c(0.6, 0.7, 0.8), 4)
#> $lower
#> [1] 0.3143726 0.3906840 0.4895897
#> 
#> $upper
#> [1] 0.3143726 0.3906840 0.4895897
#> 

## Example 3: different copula families
BiCopPar2TailDep(c(3, 4, 6), 2)
#> $lower
#> [1] 0.7071068 0.0000000 0.0000000
#> 
#> $upper
#> [1] 0.0000000 0.5857864 0.5857864
#>

No.	Lower tail dependence	Upper tail dependence
`1`	-	-
`2`	\(2t_{\nu+1}\left(-\sqrt{\nu+1}\sqrt{\frac{1-\theta}{1+\theta}}\right)\)	\(2t_{\nu+1}\left(-\sqrt{\nu+1}\sqrt{\frac{1-\theta}{1+\theta}}\right)\)
`3`	\(2^{-1/\theta}\)	-
`4`	-	\(2-2^{1/\theta}\)
`5`	-	-
`6`	-	\(2-2^{1/\theta}\)
`7`	\(2^{-1/(\theta\delta)}\)	\(2-2^{1/\delta}\)
`8`	-	\(2-2^{1/(\theta\delta)}\)
`9`	\(2^{-1/\delta}\)	\(2-2^{1/\theta}\)
`10`	-	\(2-2^{1/\theta}\) if \(\delta=1\) otherwise 0
`13`	-	\(2^{-1/\theta}\)
`14`	\(2-2^{1/\theta}\)	-
`16`	\(2-2^{1/\theta}\)	-
`17`	\(2-2^{1/\delta}\)	\(2^{-1/(\theta\delta)}\)
`18`	\(2-2^{1/(\theta\delta)}\)	-
`19`	\(2-2^{1/\theta}\)	\(2^{-1/\delta}\)
`20`	\(2-2^{1/\theta}\) if \(\delta=1\) otherwise 0	-
`23, 33`	-	-
`24, 34`	-	-
`26, 36`	-	-
`27, 37`	-	-
`28, 38`	-	-
`29, 39`	-	-
`30, 40`	-	-
`104,204`	-	\(\delta+1-(\delta^{\theta}+1)^{1/\theta}\)
`114, 214`	\(1+\delta-(\delta^{\theta}+1)^{1/\theta}\)	-
`124, 224`	-	-
`134, 234`	-	-