The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution, but this conflicts with other uses of the term: see generalized logistic distribution.
Definition The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter δ {\displaystyle \delta } . Thus if X {\displaystyle X} has a log-logistic distribution then X + δ {\displaystyle X+\delta } has a shifted log-logistic distribution. So Y {\displaystyle Y} has a shifted log-logistic distribution if log ( Y − δ ) {\displaystyle \log(Y-\delta )} has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) log-logistic. The properties of this distribution are straightforward to derive from those of the log-logistic distribution. However, an alternative parameterisation, similar to that used for the generalized Pareto distribution and the generalized extreme value distribution, gives more interpretable parameters and also aids their estimation. In this parameterisation, the cumulative distribution function (CDF) of the shifted log-logistic distribution is
F ( x ; μ , σ , ξ ) = 1 1 + ( 1 + ξ ( x − μ ) σ ) − 1 / ξ {\displaystyle F(x;\mu ,\sigma ,\xi )={\frac {1}{1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }}}}
for 1 + ξ ( x − μ ) / σ ⩾ 0 {\displaystyle 1+\xi (x-\mu )/\sigma \geqslant 0} , where μ ∈ R {\displaystyle \mu \in \mathbb {R} } is the location parameter, σ > 0 {\displaystyle \sigma >0\,} the scale parameter and ξ ∈ R {\displaystyle \xi \in \mathbb {R} } the shape parameter. Note that some references use κ = − ξ {\displaystyle \kappa =-\xi \,\!} to parameterise the shape. The probability density function (PDF) is
f ( x ; μ , σ , ξ ) = ( 1 + ξ ( x − μ ) σ ) − ( 1 / ξ + 1 ) σ [ 1 + ( 1 + ξ ( x − μ ) σ ) − 1 / ξ ] 2 , {\displaystyle f(x;\mu ,\sigma ,\xi )={\frac {\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-(1/\xi +1)}}{\sigma \left[1+\left(1+{\frac {\xi (x-\mu )}{\sigma }}\right)^{-1/\xi }\right]^{2}}},}
again, for 1 + ξ ( x − μ ) / σ ⩾ 0. {\displaystyle 1+\xi (x-\mu )/\sigma \geqslant 0.}
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