Based on the distribution function of the standard logistic distribution.
\[g(\mu)=\frac{\mu}{1-\mu}\]
\[g^{-1}(\eta)=\frac{\exp(\eta)}{1+\exp(\eta)}\]
Based on the distribution function of the standard normal distribution.
\[g(\mu)=\Phi^{-1}(\mu)\]
\[g^{-1}(\eta)=\Phi(\eta)\]
Based on the distribution function of the standard Cauchy distribution.
\[g(\mu)=\frac{1}{\pi}\arctan(\mu)+\frac{1}{2}\]
\[g^{-1}(\eta)=\tan[\pi(\eta-\frac{1}{2})]\]
Related to the distribution function of the extreme value (Gumbel) distribution. The link is not symmetric.
\[\quad g(\mu)=\log(-\log(1-\mu))\]
\[g^{-1}(\eta)=1-\exp(-\exp(\eta))\]
