BiCopTau2Par: Parameter of a Bivariate Copula for a given Kendall's Tau Value

Description

This function computes the parameter of a (one parameter) bivariate copula for a given value of Kendall's tau.

Usage

BiCopTau2Par(family, tau)

Arguments

family

An integer defining the bivariate copula family: 0 = independence copula 1 = Gaussian copula 2 = Student t copula (Here only the first parameter can be computed) 3 = Clayton copula 4 = G

tau

Kendall's tau value (vector with elements in [-1,1]).

Value

Parameter (vector) corresponding to the bivariate copula family and the value(s) of Kendall's tau ($\tau$). ll{ No. (family) Parameter (par) 1, 2 $\sin(\tau \frac{\pi}{2})$ 3, 13 $2\frac{\tau}{1-\tau}$ 4, 14 $\frac{1}{1-\tau}$ 5 no closed form expression (numerical inversion) 6, 16 no closed form expression (numerical inversion) 23, 33 $2\frac{\tau}{1+\tau}$ 24, 34 $-\frac{1}{1+\tau}$ 26, 36 no closed form expression (numerical inversion) }

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
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:
vtau <- seq(from = 0.1, to = 0.8, length.out = 100)
thetaC <- BiCopTau2Par(family = 3, tau = vtau)
thetaG <- BiCopTau2Par(family = 4, tau = vtau)
thetaF <- BiCopTau2Par(family = 5, tau = vtau)
thetaJ <- BiCopTau2Par(family = 6, tau = vtau)
plot(thetaC ~ vtau, type = "l", ylim = range(thetaF))
lines(thetaG ~ vtau, col = 2)
lines(thetaF ~ vtau, col = 3)
lines(thetaJ ~ vtau, col = 4)

# Test BiCopTau2Par
BiCopTau2Par(family = 0, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 1, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 2, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 3, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 4, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 5, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 6, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 13, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 14, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 16, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 23, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 24, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 26, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 33, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 34, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 36, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 41, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 51, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 61, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 71, tau = c(0.4,0.5,0.6))
BiCopTau2Par(family = 41, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 51, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 61, tau = -c(0.4,0.5,0.6))
BiCopTau2Par(family = 71, tau = -c(0.4,0.5,0.6))