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)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'')
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)
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)
numeric; single number or vector of size n; copula parameter.
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.
BiCop object containing the family and parameter
specification.
logical; default is TRUE; if FALSE, checks
for family/parameter-consistency are omitted (should only be used with
care).
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 tail dependence coefficient for the given bivariate
copula family family and parameter(s) par, par2:
$$ \lambda_U = \lim_{u\nearrow 1}\frac{1-2u+C(u,u)}{1-u} $$
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 - -
If the family and parameter specification is stored in a BiCop object
obj, the alternative version
BiCopPar2TailDep(obj)
can be used.
Joe, H. (1997). Multivariate Models and Dependence Concepts. Chapman and Hall, London.
# NOT RUN {
## 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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