Table 1 The \(\vartheta \left(x\right)\) functions and their extended copulas and extended co-copulas.
S. no | \(\vartheta \left(x\right)\) | \(C\left(x,y\right)\) | \({C}^{*}\langle x,y\rangle\) |
|---|---|---|---|
1 | \({\left(-\mathrm{ln}(x)\right)}^{\lambda }\) | \(xy\) | \(x+y-xy\) |
2 | \({\left(x\right)}^{-\lambda }-1\) | \(\left({\left(x\right)}^{-\lambda }+{\left(y\right)}^{-\lambda }-1\right)\) | \({\left(1-x\right)}^{-\lambda }+{\left(1-x\right)}^{-\lambda }-xy\) |
3 | \(ln\left(\frac{{e}^{-\lambda x}-1}{{e}^{-\lambda }-1}\right)\) | \(\left(-\frac{1}{\lambda }\right)ln\left(\frac{\left({e}^{-\lambda x}-1\right)\left({e}^{-\lambda x}-1\right)}{{e}^{-\lambda }-1}+1\right)\) | \(1+\frac{1}{\lambda }ln\left(\frac{\left({e}^{-\lambda x}-1\right)\left({e}^{-\lambda x}-1\right)}{{e}^{-\lambda }-1}+1\right)\) |
4 | \(ln\left(\frac{1-\lambda (1-x)}{x}\right)\) | \(\left(\frac{xy}{\left(1-\lambda \right)\left(1-x\right)\left(1-y\right)}\right)\) | \(1-\left(\frac{\left(1-x\right)\left(1-y\right)}{\left(1-\lambda xy\right)}\right)\) |
5 | \(-ln\left(1-{\left(1-x\right)}^{\lambda }\right)\) | \(1-{\left({\left(1-x\right)}^{\lambda }+{\left(1-y\right)}^{\lambda }-{\left(1-x\right)}^{\lambda }{\left(1-y\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}\) | \({\left({\left(x\right)}^{\lambda }+{\left(y\right)}^{\lambda }-{\left(x\right)}^{\lambda }{\left(y\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}\) |
