The parametric generator functions of copula functions are list below:
The Clayton copula has a generator $$\phi_{\eta}(t) = (1+t)^{-1/\eta},$$
with \(\eta > 0\) and Kendall's \(\tau = \eta/(2+\eta)\).
The Gumbel copula has a generator $$\phi_{\eta}(t) = \exp(-t^{1/\eta}),$$
with \(\eta \geq 1\) and Kendall's \(\tau = 1 - 1/\eta\).
The Frank copula has a generator $$\phi_{\eta}(t) = -\eta^{-1}\log \{1+e^{-t}(e^{-\eta}-1)\},$$
with \(\eta \geq 0\) and Kendall's \(\tau = 1+4\{D_1(\eta)-1\}/\eta\),
in which \(D_1(\eta) = \frac{1}{\eta} \int_{0}^{\eta} \frac{t}{e^t-1}dt\).
The AMH copula has a generator $$\phi_{\eta}(t) = (1-\eta)/(e^{t}-\eta),$$
with \(\eta \in [0,1)\) and Kendall's \(\tau = 1-2\{(1-\eta)^2 \log (1-\eta) + \eta\}/(3\eta^2)\).
The Joe copula has a generator $$\phi_{\eta}(t) = 1-(1-e^{-t})^{1/\eta},$$
with \(\eta \geq 1\) and Kendall's \(\tau = 1 - 4 \sum_{k=1}^{\infty} \frac{1}{k(\eta k+2)\{\eta(k-1)+2\}}\).
The marginal survival distributions are listed below:
The Weibull (PH) survival distribution is $$\exp \{-(t/\lambda)^k e^{Z^{\top}\beta}\},$$
with \(\lambda > 0\) as scale and \(k > 0\) as shape.
The Gompertz (PH) survival distribution is $$\exp \{-\frac{b}{a}(e^{at}-1) e^{Z^{\top}\beta}\},$$
with \(a > 0\) as shape and \(b > 0\) as rate
The Loglogistic (PO) survival distribution is $$\{1+(t/\lambda)^{k} e^{Z^{\top}\beta} \}^{-1},$$
with \(\lambda > 0\) as scale and \(k > 0\) as shape.