Theoretical value of Kendall's tau (vector) corresponding to the
bivariate copula family and parameter vector \((\theta, \delta) =\)
(par, par2).
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 |
\(1+4\int_0^1 ( (1-(1-t)^{\theta})^{-\delta} - 1)
/( -\theta\delta(1-t)^{\theta-1}(1-(1-t)^{\theta})^{-\delta-1} ) dt\) |
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 |
\(-1-4\int_0^1 ( (1-(1-t)^{-\theta})^{\delta} - 1)
/( -\theta\delta(1-t)^{-\theta-1}(1-(1-t)^{-\theta})^{\delta-1} ) dt\) |
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 \(A(t) = (1-\delta)(1-t)+[(1-t)^{-\theta}+(\delta t)^{-\theta}]^{-1/\theta}\) |