VineCopula (version 1.6-1)

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, obj = NULL)

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

par

Copula parameter.

par2

Second parameter for the two parameter t-, BB1, BB6, BB7, BB8, Tawn type 1 and type 2 copulas (default: par2 = 0).

obj

BiCop object containing the family and parameter specification.

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. lll{ 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.

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)