Kendall's Tau Value of a Bivariate Copula
This function computes the theoretical Kendall's tau value of a bivariate copula for given parameter values.
BiCopPar2Tau(family, par, par2 = 0, obj = NULL, check.pars = TRUE)
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 |
If the family and parameter specification is stored in a BiCop()
object obj, the alternative version
BiCopPar2Tau(obj)
can be used.
Theoretical value of Kendall's tau (vector) corresponding to the
bivariate copula family and parameter vector (θ, δ) =
(par, par2).
No. (family) |
Kendall's tau (tau) |
1, 2 |
2 / π arcsin(θ) |
3, 13 |
θ / (θ+2) |
4, 14 |
1-1/θ |
5 |
1-4/θ + 4 D_1(θ)/θ |
| with D_1(θ)= \int_0^θ (x/θ)/(exp(x)-1)dx (Debye function) | |
6, 16 |
1+4/θ^2\int_0^1 x\log(x)(1-x)^{2(1-θ)/θ}dx |
7, 17 |
1-2/(δ(θ+2)) |
8, 18 |
1+4\int_0^1 -\log(-(1-t)^θ+1) (1-t-(1-t)^{-θ}+(1-t)^{-θ}t)/(δθ) dt |
9, 19 |
1+4\int_0^1 ( (1-(1-t)^{θ})^{-δ} - 1) /( -θδ(1-t)^{θ-1}(1-(1-t)^{θ})^{-δ-1} ) dt |
10, 20 |
1+4\int_0^1 -\log ≤ft(((1-tδ)^θ-1)/((1-δ)^θ-1) \right) |
| * (1-tδ-(1-tδ)^{-θ}+(1-tδ)^{-θ}tδ)/(θδ) dt | |
23, 33 |
θ/(2-θ) |
24, 34 |
-1-1/θ |
26, 36 |
-1-4/θ^2 \int_0^1 x\log(x)(1-x)^{-2(1+θ)/θ}dx |
27, 37 |
1-2/(δ(θ+2)) |
28, 38 |
-1-4\int_0^1 -\log(-(1-t)^{-θ}+1) (1-t-(1-t)^{θ}+(1-t)^{θ}t)/(δθ) dt |
29, 39 |
-1-4\int_0^1 ( (1-(1-t)^{-θ})^{δ} - 1) /( -θδ(1-t)^{-θ-1}(1-(1-t)^{-θ})^{δ-1} ) dt |
30, 40 |
-1-4\int_0^1 -\log ≤ft( ((1+tδ)^{-θ}-1)/((1+δ)^{-θ}-1) \right) |
| * (1+tδ-(1+tδ)^{θ}-(1+tδ)^{θ}tδ)/(θδ) dt | |
104,114 |
\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
| with A(t) = (1-δ)t+[(δ(1-t))^{θ}+t^{θ}]^{1/θ} | |
204,214 |
\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
| with A(t) = (1-δ)(1-t)+[(1-t)^{-θ}+(δ t)^{-θ}]^{-1/θ} | |
124,134 |
-\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
| with A(t) = (1-δ)t+[(δ(1-t))^{-θ}+t^{-θ}]^{-1/θ} | |
224,234 |
-\int_0^1 \frac{t(1-t)A^{\prime\prime}(t)}{A(t)}dt |
| with A(t) = (1-δ)(1-t)+[(1-t)^{-θ}+(δ t)^{-θ}]^{-1/θ} | |
The number n can be chosen arbitrarily, but must agree across
arguments.
Ulf Schepsmeier, Tobias Erhardt
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.
## 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)
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