VineCopula (version 2.6.1)

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, check.pars = TRUE)

Value

lower

Lower tail dependence coefficient for the given bivariate copula family and parameter(s) par, par2: $${\lambda }_L=\underset{u\searrow 0}{lim}\frac{C(u,u)}{u}$$

upper

Upper tail dependence coefficient for the given bivariate copula family family and parameter(s) par, par2: $${\lambda }_U=\underset{u\nearrow 1}{lim}\frac{1-2u+C(u,u)}{1-u}$$

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	$2t_{\nu +1}(-\sqrt{\nu +1}\sqrt{\frac{1-\theta }{1+\theta }})$	$2t_{\nu +1}(-\sqrt{\nu +1}\sqrt{\frac{1-\theta }{1+\theta }})$
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	-	-
104,204	-	$\delta +1-({\delta }^{\theta }+1{)}^{1/\theta }$
114, 214	$1+\delta -({\delta }^{\theta }+1{)}^{1/\theta }$	-
124, 224	-	-
134, 234	-	-

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).

Author

Eike Brechmann

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)
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)