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.
\delta
X
X+\delta
Y
log(Y-\delta)
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;\mu,\sigma,\xi)=
1 | |||||
|
1+\xi(x-\mu)/\sigma\geqslant0
\mu\inR
\sigma>0
\xi\inR
\kappa=-\xi
The probability density function (PDF) is
f(x;\mu,\sigma,\xi)=
| |||||
|
,
1+\xi(x-\mu)/\sigma\geqslant0.
The shape parameter
\xi
|\xi|>1
x=\mu-\sigma/\xi
\xi
x=\mu.
\mu=\sigma/\xi,
\xi
\xi=1
\xi=1.
The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.
An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter
\alpha
\beta
\gamma
f(x)=
\alpha | ( | |
\beta |
x-\gamma | |
\beta |
)\alpha(1+(
x-\gamma | |
\beta |
)\alpha)-2
The CDF is given by
F(x)=(1+(
\beta | |
x-\gamma |
)\alpha)-1
The mean is
\beta\theta\csc(\theta)+\gamma
\beta2\theta[2\csc(2\theta)-\theta\csc2(\theta)]
\theta=
\pi | |
\alpha |