Table 1 Copulas functions, parameters, Kendall’s τ , and tail dependence
From: Price asymmetry between different pork cuts in the USA: a copula approach
Copulas | Parameters | Kendall’s τ | Tail dependence |
|---|---|---|---|
( λ L , λ U ) | |||
Gaussian | ρ∈(−1,1) | \(\frac {2}{\pi }\)arcsin(ρ) | (0,0) |
S t u d e n t−t | ρ∈(−1,1) | \(\frac {2}{\pi }\)arcsin(ρ) | \(2t_{\nu +1}\left (-\sqrt {\nu +1} \sqrt {\frac {1-\rho }{1+\rho }}\right)\), |
(for ν > 2, where ν | \(2t_{\nu +1}\left (-\sqrt {\nu +1} \sqrt {\frac {1-\rho }{1+\rho }}\right)\) | ||
are the degrees of freedom) | |||
Clayton | θ>0 | \(\frac {\theta }{\theta +2}\) | \(\left (2^{\frac {-1}{\,\theta }}, 0\right)\) |
Gumbel | θ≥1 | 1- \(\frac {1}{\theta }\) | \(\left (0, 2 - 2^{\frac {1}{\theta }}\right)\) |
Frank | θ∈R∖{0} | 1 -\(\frac {4}{\theta }+4\frac {D(\theta)}{\theta }\) | (0,0) |
Joe | θ≥1 | 1+\(\frac {4}{\theta ^{2}} {\int _{0}^{1}} t log(t) (1-t)^{2 (1-\theta)/{\theta }} \,dt\) | \(\left (0, 2 - 2^{\frac {1}{\theta }}\right)\) |
C l a y t o n−G u m b e l | θ 1>0,θ 2≥1 | 1 - \(\frac {2}{\theta _{2}(\theta _{1}+2)}\) | \(\left (2^{\frac {-1}{{\theta _{1}\theta _{2}}}}, 2 - 2^{\frac {1}{\theta _{2}}}\right)\) |
J o e−C l a y t o n | θ 1≥1, θ 2>0 | 1+\(\frac {4}{\theta _{1}\theta _{2}} {\int _{0}^{1}}\left (\vphantom {\frac {(1-(1-t)^{\theta _{1}})^{-\theta _{2}}- 1}{(1-t)^{\theta _{2}-1}}}-(1-(1-t)^{\theta _{1}})^{\theta _{2}+1}\right.\) | \(\left (2^{\frac {-1}{\theta _{2}}}, 2 - 2^{\frac {1}{{\theta _{1}}}}\right)\) |
*\(\left.\frac {({1-(1-t)}^{\theta _{1}})^{{-\theta }_{2}}- 1}{(1-t)^{{\theta }_{2}-1}}\right)dt\) |