CDVine (version 1.4)

BiCopPar2TailDep: Tail dependence coefficients of a bivariate copula

Description

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

Usage

BiCopPar2TailDep(family, par, par2=0)

Arguments

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).

Value

lower

Lower tail dependence coefficient of the given bivariate copula family $C$: $$ \lambda_L = \lim_{u\searrow 0}\frac{C(u,u)}{u} $$

upper

Upper tail dependence coefficient of the given bivariate copula family $C$: $$ \lambda_U = \lim_{u\nearrow 1}\frac{1-2u+C(u,u)}{1-u} $$

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	$2t_{\nu+1}(-\sqrt{\nu+1}\sqrt{(1-\theta)/(1+\theta)})$
$2t_{\nu+1}(-\sqrt{\nu+1}\sqrt{(1-\theta)/(1+\theta)})$	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	-
-	No.

References

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

See Also

BiCopPar2Tau

Examples

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

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