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, check.taus = TRUE)

Arguments

family

integer; single number or vector of size n; defines 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 = Gumbel copula 5 = Frank copula 6 = Joe 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'') 23 = rotated Clayton copula (90 degrees) 24 = rotated Gumbel copula (90 degrees) 26 = rotated Joe copula (90 degrees) 33 = rotated Clayton copula (270 degrees) 34 = rotated Gumbel copula (270 degrees) 36 = rotated Joe copula (270 degrees) Note that (with exception of the t-copula) two parameter bivariate copula families cannot be used.

tau

numeric; single number or vector of size n; Kendall's tau value (vector with elements in [-1,1]).

check.taus

logical; default is TRUE; if FALSE, checks for family/tau-consistency are omitted (should only be used with care).

Value

Parameter (vector) corresponding to the bivariate copula family and the value(s) of Kendall's tau (\(\tau\)).

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`	\(-\frac{1}{1+\tau}\)

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

# NOT RUN {
## Example 1: Gaussian copula
tau0 <- 0.5
rho <- BiCopTau2Par(family = 1, tau = tau0)
BiCop(1, tau = tau0)$par  # alternative

## Example 2:
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)

## 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)

# }
# NOT RUN {
# }