Based on the distribution function of the standard logistic distribution.
g(μ)=μ1−μ
g−1(η)=exp(η)1+exp(η)
Based on the distribution function of the standard normal distribution.
g(μ)=Φ−1(μ)
g−1(η)=Φ(η)
Based on the distribution function of the standard Cauchy distribution.
g(μ)=1πarctan(μ)+12
g−1(η)=tan[π(η−12)]
Related to the distribution function of the extreme value (Gumbel) distribution. The link is not symmetric.
g(μ)=log(−log(1−μ))
g−1(η)=1−exp(−exp(η))