BiCopPar2Tau {CDVine} | R Documentation |
This function computes the theoretical Kendall's tau value of a bivariate copula for given parameter values.
BiCopPar2Tau(family, par, par2=0)
family |
An integer defining the bivariate copula family: |
par |
Copula parameter. |
par2 |
Second parameter for the two parameter BB1, BB6, BB7 and BB8 copulas (default: |
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 | |
Ulf Schepsmeier
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.
## Example 1: Gaussian copula tt1 = BiCopPar2Tau(1,0.7) # transform back BiCopTau2Par(1,tt1) ## Example 2: Clayton copula BiCopPar2Tau(3,1.3)