BiCopTau2Par {VineCopula}

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'') \cr `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}
⁠26, 36⁠	no closed form expression (numerical inversion)

Note

The number n can be chosen arbitrarily, but must agree across arguments.

Author(s)

Jakob Stoeber, Eike Brechmann, 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

BiCopPar2Tau()

Examples

## 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)