| BiCopPar2Tau {VineCopula} | R Documentation |
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, obj = NULL, check.pars = TRUE)
Arguments
family |
integer; single number or vector of size |
par |
numeric; single number or vector of size |
par2 |
numeric; single number or vector of size |
obj |
|
check.pars |
logical; default is |
Details
If the family and parameter specification is stored in a BiCop()
object obj, the alternative version
BiCopPar2Tau(obj)
can be used.
Value
Theoretical value of Kendall's tau (vector) corresponding to the
bivariate copula family and parameter vector (\theta, \delta) =
(par, par2).
No. (family) | Kendall's tau (tau) |
1, 2 | \frac{2}{\pi}\arcsin(\theta) |
3, 13 | \frac{\theta}{\theta+2} |
4, 14 | 1-\frac{1}{\theta} |
5 | 1-\frac{4}{\theta}+4\frac{D_1(\theta)}{\theta} |
with D_1(\theta)=\int_0^\theta \frac{x/\theta}{\exp(x)-1}dx (Debye function) |
|
6, 16 | 1+\frac{4}{\theta^2}\int_0^1
x\log(x)(1-x)^{2(1-\theta)/\theta}dx |
7, 17 | 1-\frac{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} - 1)
/( -\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 | \frac{\theta}{2-\theta} |
24, 34 | -1-\frac{1}{\theta} |
26, 36 | -1-\frac{4}{\theta^2}\int_0^1
x\log(x)(1-x)^{-2(1+\theta)/\theta}dx |
27, 37 | -1-\frac{2}{\delta(2-\theta)} |
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} - 1)
/( -\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 |
|
104,114 | \int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
with A(t) = (1-\delta)t+[(\delta(1-t))^{\theta}+t^{\theta}]^{1/\theta} |
|
204,214 | \int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
with A(t) = (1-\delta)(1-t)+[(1-t)^{-\theta}+(\delta t)^{-\theta}]^{-1/\theta} |
|
124,134 | -\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
with A(t) = (1-\delta)t+[(\delta(1-t))^{-\theta}+t^{-\theta}]^{-1/\theta} |
|
224,234 | -\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
with A(t) = (1-\delta)(1-t)+[(1-t)^{-\theta}+(\delta t)^{-\theta}]^{-1/\theta} |
|
Note
The number n can be chosen arbitrarily, but must agree across
arguments.
Author(s)
Ulf Schepsmeier, Tobias Erhardt
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
Examples
## Example 1: Gaussian copula
tau0 <- 0.5
rho <- BiCopTau2Par(family = 1, tau = tau0)
# transform back
tau <- BiCopPar2Tau(family = 1, par = rho)
tau - 2/pi*asin(rho)
## Example 2:
vpar <- seq(from = 1.1, to = 10, length.out = 100)
tauC <- BiCopPar2Tau(family = 3, par = vpar)
tauG <- BiCopPar2Tau(family = 4, par = vpar)
tauF <- BiCopPar2Tau(family = 5, par = vpar)
tauJ <- BiCopPar2Tau(family = 6, par = vpar)
plot(tauC ~ vpar, type = "l", ylim = c(0,1))
lines(tauG ~ vpar, col = 2)
lines(tauF ~ vpar, col = 3)
lines(tauJ ~ vpar, col = 4)
## Example 3: different copula families
theta <- BiCopTau2Par(family = c(3,4,6), tau = c(0.4, 0.5, 0.6))
BiCopPar2Tau(family = c(3,4,6), par = theta)