VineCopula (version 1.4)

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

par

Copula parameter (vector).

par2

Second parameter (vector of same length as par) for the two parameter t-, BB1, BB6, BB7, BB8, Tawn type 1 and type 2 copulas (default: par2 = 0). Note that the degrees of freedom parameter of the t-copula does not need to be

Value

• Theoretical value of Kendall's tau (vector) corresponding to the bivariate copula family and parameter(vectors) ($\theta$ for one parameter families and the first parameter of the t-copula, $\theta$ and $\delta$ for the two parameter BB1, BB6, BB7, BB8, Tawn type 1 and type 2 copulas). ll{ 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 $1+4\int_0^1 ( (1-(1-t)^{\theta})^{-\delta} - )/( -\theta\delta(1-t)^{\theta-1}(1-(1-t)^{\theta})^{-\delta-1} ) dt$ 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 $-1-4\int_0^1 ( (1-(1-t)^{-\theta})^{\delta} - )/( -\theta\delta(1-t)^{-\theta-1}(1-(1-t)^{-\theta})^{\delta-1} ) dt$ 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 $A(t) = (1-\delta)(1-t)+[(1-t)^{-\theta}+(\delta t)^{-\theta}]^{-1/\theta}$ }

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

BiCopTau2Par

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: Clayton copula
theta <- BiCopTau2Par(family = 3, tau = c(0.4, 0.5, 0.6))
BiCopPar2Tau(family = 3, par = theta)


## Example 3:
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)

# Test BiCopPar2Tau (one parametric families)
theta <- BiCopTau2Par(family = 0, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 0, par = theta)
theta <- BiCopTau2Par(family = 1, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 1, par = theta)
theta <- BiCopTau2Par(family = 3, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 3, par = theta)
theta <- BiCopTau2Par(family = 4, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 4, par = theta)
theta <- BiCopTau2Par(family = 5, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 5, par = theta)
theta <- BiCopTau2Par(family = 6, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 6, par = theta)
theta <- BiCopTau2Par(family = 13, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 13, par = theta)
theta <- BiCopTau2Par(family = 14, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 14, par = theta)
theta <- BiCopTau2Par(family = 16, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 16, par = theta)
theta <- BiCopTau2Par(family = 23, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 23, par = theta)
theta <- BiCopTau2Par(family = 24, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 24, par = theta)
theta <- BiCopTau2Par(family = 26, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 26, par = theta)
theta <- BiCopTau2Par(family = 33, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 33, par = theta)
theta <- BiCopTau2Par(family = 34, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 34, par = theta)
theta <- BiCopTau2Par(family = 36, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 36, par = theta)
theta <- BiCopTau2Par(family = 41, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 41, par = theta)
theta <- BiCopTau2Par(family = 51, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 51, par = theta)
theta <- BiCopTau2Par(family = 61, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 61, par = theta)
theta <- BiCopTau2Par(family = 71, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 71, par = theta)
theta <- BiCopTau2Par(family = 41, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 41, par = theta)
theta <- BiCopTau2Par(family = 51, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 51, par = theta)
theta <- BiCopTau2Par(family = 61, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 61, par = theta)
theta <- BiCopTau2Par(family = 71, tau = -c(0.4,0.5,0.6))
BiCopPar2Tau(family = 71, par = theta)

# Test BiCopPar2Tau (two parametric families)
theta <- BiCopTau2Par(family = 2, tau = c(0.4,0.5,0.6))
BiCopPar2Tau(family = 2, par = theta)
theta <- 1:3
delta <- 1:3
BiCopPar2Tau(family = 7, par = theta, par2 = delta)
BiCopPar2Tau(family = 17, par = theta, par2 = delta)
theta <- -(1:3)
delta <- -(1:3)
BiCopPar2Tau(family = 27, par = theta, par2 = delta)
BiCopPar2Tau(family = 37, par = theta, par2 = delta)
theta <- 2:4
delta <- 1:3
BiCopPar2Tau(family = 8, par = theta, par2 = delta)
BiCopPar2Tau(family = 18, par = theta, par2 = delta)
theta <- -(2:4)
delta <- -(1:3)
BiCopPar2Tau(family = 28, par = theta, par2 = delta)
BiCopPar2Tau(family = 38, par = theta, par2 = delta)
theta <- 1:3
delta <- 1:3
BiCopPar2Tau(family = 9, par = theta, par2 = delta)
BiCopPar2Tau(family = 19, par = theta, par2 = delta)
theta <- -(1:3)
delta <- -(1:3)
BiCopPar2Tau(family = 29, par = theta, par2 = delta)
BiCopPar2Tau(family = 39, par = theta, par2 = delta)
theta <- 2:4
delta <- c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 10, par = theta, par2 = delta)
BiCopPar2Tau(family = 20, par = theta, par2 = delta)
theta <- -(2:4)
delta <- -c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 30, par = theta, par2 = delta)
BiCopPar2Tau(family = 40, par = theta, par2 = delta)

theta <- 2:4
delta <- c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 104, par = theta, par2 = delta)
BiCopPar2Tau(family = 114, par = theta, par2 = delta)
theta <- -(2:4)
delta <- c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 124, par = theta, par2 = delta)
BiCopPar2Tau(family = 134, par = theta, par2 = delta)

theta <- 2:4
delta <- c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 204, par = theta, par2 = delta)
BiCopPar2Tau(family = 214, par = theta, par2 = delta)
theta <- -(2:4)
delta <- c(0.1, 0.5, 0.9)
BiCopPar2Tau(family = 224, par = theta, par2 = delta)
BiCopPar2Tau(family = 234, par = theta, par2 = delta)