BiCopPar2Tau: Kendall's tau value of a bivariate copula

In CDVine: Statistical Inference of C- And D-Vine Copulas

Description

This function computes the theoretical Kendall's tau value of a bivariate copula for given parameter values.

Usage

BiCopPar2Tau(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 BB1, BB6, BB7 and BB8 copulas (default: par2 = 0). Note that the degrees of freedom parameter of the t-copula does not need to be set, because the theoretical Kendall's tau value of the t-copula is independent of this choice.

Value

Theoretical value of Kendall's tau corresponding to the bivariate copula family and parameter(s) (θ for one parameter families and the first parameter of the t-copula, θ and δ for the two parameter BB1, BB6, BB7 and BB8 copulas).

No.	Kendall's tau
1, 2	2 / π arcsin(θ)
3, 13	θ / (θ+2)
4, 14	1-1/θ
5	1-4/θ + 4 D_1(θ)/θ
	with D_1(θ)=\int_0^θ (x/θ)/(exp(x)-1)dx (Debye function)
6, 16	1+4/θ^2\int_0^1 x\log(x)(1-x)^{2(1-θ)/θ}dx
7, 17	1-2/(δ(θ+2))
8, 18	1+4\int_0^1 -\log(-(1-t)^θ+1)(1-t-(1-t)^{-θ}+(1-t)^{-θ}t)/(δθ) dt
9, 19	1+4\int_0^1 ( (1-(1-t)^{θ})^{-δ} - )/( -θδ(1-t)^{θ-1}(1-(1-t)^{θ})^{-δ-1} ) dt
10, 20	1+4\int_0^1 -\log ≤ft( ((1-tδ)^θ-1)/((1-δ)^θ-1) \right)
	* (1-tδ-(1-tδ)^{-θ}+(1-tδ)^{-θ}tδ)/(θδ) dt
23, 33	θ/(2-θ)
24, 34	-1-1/θ
26, 36	-1-4/θ^2\int_0^1 x\log(x)(1-x)^{-2(1+θ)/θ}dx
27, 37	1-2/(δ(θ+2))
28, 38	-1-4\int_0^1 -\log(-(1-t)^{-θ}+1)(1-t-(1-t)^{θ}+(1-t)^{θ}t)/(δθ) dt
29, 39	-1-4\int_0^1 ( (1-(1-t)^{-θ})^{δ} - )/( -θδ(1-t)^{-θ-1}(1-(1-t)^{-θ})^{δ-1} ) dt
30, 40	-1-4\int_0^1 -\log ≤ft( ((1+tδ)^{-θ}-1)/((1+δ)^{-θ}-1) \right)
	* (1+tδ-(1+tδ)^{θ}-(1+tδ)^{θ}tδ)/(θδ) dt

Author(s)

Ulf Schepsmeier

References

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

Czado, C., U. Schepsmeier, and A. Min (2012). Maximum likelihood estimation of mixed C-vines with application to exchange rates. Statistical Modelling, 12(3), 229-255.

See Also

CDVinePar2Tau, BiCopTau2Par

Examples

## Example 1: Gaussian copula
tt1 = BiCopPar2Tau(1,0.7)

# transform back
BiCopTau2Par(1,tt1)


## Example 2: Clayton copula
BiCopPar2Tau(3,1.3)

CDVine documentation built on May 2, 2019, 9:28 a.m.