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)
family |
integer; single number or vector of size |
par |
numeric; single number or vector of size |
par2 |
numeric; single number or vector of size |
obj |
|
check.pars |
logical; default is |
If the family and parameter specification is stored in a BiCop object
obj, the alternative version
BiCopPar2TailDep(obj)
can be used.
lower |
Lower tail dependence coefficient for the given
bivariate copula λ_L = lim_{u->0} C(u,u)/u |
upper |
Upper tail dependence coefficient for the given bivariate
copula family 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 |
- | - |
104,204 |
- | δ+1-(δ^{θ}+1)^{1/θ} |
114, 214 |
1+δ-(δ^{θ}+1)^{1/θ} | - |
124, 224 |
- | - |
134, 234 |
- | - |
The number n can be chosen arbitrarily, but must agree across
arguments.
Eike Brechmann
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
## Example 1: Gaussian copula BiCopPar2TailDep(1, 0.7) BiCop(1, 0.7)$taildep # alternative ## Example 2: Student-t copula BiCopPar2TailDep(2, c(0.6, 0.7, 0.8), 4) ## Example 3: different copula families BiCopPar2TailDep(c(3, 4, 6), 2)
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