![]() | ||
Parameters μ ∈ ( − ∞ , + ∞ ) {\displaystyle \mu \in (-\infty ,+\infty )\,} location (real) σ ∈ ( 0 , + ∞ ) {\displaystyle \sigma \in (0,+\infty )\,} scale (real) ξ ∈ ( − ∞ , + ∞ ) {\displaystyle \xi \in (-\infty ,+\infty )\,} shape (real) Support x ⩾ μ − σ / ξ ( ξ > 0 ) {\displaystyle x\geqslant \mu -\sigma /\xi \,\;(\xi >0)} x ⩽ μ − σ / ξ ( ξ < 0 ) {\displaystyle x\leqslant \mu -\sigma /\xi \,\;(\xi <0)} x ∈ ( − ∞ , + ∞ ) ( ξ = 0 ) {\displaystyle x\in (-\infty ,+\infty )\,\;(\xi =0)} PDF ( 1 + ξ z ) − ( 1 / ξ + 1 ) σ ( 1 + ( 1 + ξ z ) − 1 / ξ ) 2 {\displaystyle {\frac {(1+\xi z)^{-(1/\xi +1)}}{\sigma \left(1+(1+\xi z)^{-1/\xi }\right)^{2}}}} where z = ( x − μ ) / σ {\displaystyle z=(x-\mu )/\sigma \,} CDF ( 1 + ( 1 + ξ z ) − 1 / ξ ) − 1 {\displaystyle \left(1+(1+\xi z)^{-1/\xi }\right)^{-1}\,} where z = ( x − μ ) / σ {\displaystyle z=(x-\mu )/\sigma \,} Mean μ + σ ξ ( α csc ( α ) − 1 ) {\displaystyle \mu +{\frac {\sigma }{\xi }}(\alpha \csc(\alpha )-1)} where α = π ξ {\displaystyle \alpha =\pi \xi \,} Median μ {\displaystyle \mu \,} |
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.
Contents
Definition
The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter
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
for
The probability density function (PDF) is
again, for
The shape parameter
Related distributions
Applications
The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.
Alternate parameterization
An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter
The CDF is given by
The mean is