BiCopPar2Tau: Kendall's tau value of a bivariate copula

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) ($\theta$ for one parameter families and the first parameter of the t-copula, $\theta$ and $\delta$ for the two parameter BB1, BB6, BB7 and BB8 copulas).

No.

Kendall's tau

1, 2

$2 / \pi arcsin(\theta)$

3, 13

$\theta / (\theta+2)$

4, 14

$1-1/\theta$

5

$1-4/\theta + 4 D_1(\theta)/\theta$

with $D_1(\theta)=\int_0^\theta (x/\theta)/(exp(x)-1)dx$ (Debye function)

6, 16

$1+4/\theta^2\int_0^1 x\log(x)(1-x)^{2(1-\theta)/\theta}dx$

7, 17

$1-2/(\delta(\theta+2))$

8, 18

$1+4\int_0^1 -\log(-(1-t)^\theta+1)(1-t-(1-t)^{-\theta}+(1-t)^{-\theta}t)/(\delta\theta) dt$

9, 19

$1+4\int_0^1 ( (1-(1-t)^{\theta})^{-\delta} - )/( -\theta\delta(1-t)^{\theta-1}(1-(1-t)^{\theta})^{-\delta-1} ) dt$

10, 20

$1+4\int_0^1 -\log \left( ((1-t\delta)^\theta-1)/((1-\delta)^\theta-1) \right) $

$* (1-t\delta-(1-t\delta)^{-\theta}+(1-t\delta)^{-\theta}t\delta)/(\theta\delta) dt$

23, 33

$\theta/(2-\theta)$

24, 34

$-1-1/\theta$

26, 36

$-1-4/\theta^2\int_0^1 x\log(x)(1-x)^{-2(1+\theta)/\theta}dx$

27, 37

$1-2/(\delta(\theta+2))$

28, 38

$-1-4\int_0^1 -\log(-(1-t)^{-\theta}+1)(1-t-(1-t)^{\theta}+(1-t)^{\theta}t)/(\delta\theta) dt$

29, 39

$-1-4\int_0^1 ( (1-(1-t)^{-\theta})^{\delta} - )/( -\theta\delta(1-t)^{-\theta-1}(1-(1-t)^{-\theta})^{\delta-1} ) dt$

30, 40

$-1-4\int_0^1 -\log \left( ((1+t\delta)^{-\theta}-1)/((1+\delta)^{-\theta}-1) \right)$

$* (1+t\delta-(1+t\delta)^{\theta}-(1+t\delta)^{\theta}t\delta)/(\theta\delta) dt$

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.

Examples

Run this code

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

# transform back
BiCopTau2Par(1,tt1)


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