Shifted log-logistic distribution

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Shifted log-logistic
Probability density function
values of as shown in legend
Cumulative distribution function
values of as shown in legend

location (real)
scale (real)

shape (real)






The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution.[1][2] It has also been called the generalized logistic distribution,[3] but this conflicts with other uses of the term: see generalized logistic distribution.


The shifted log-logistic distribution can be obtained from the log-logistic distribution by addition of a shift parameter . Thus if has a log-logistic distribution then has a shifted log-logistic distribution. So has a shifted log-logistic distribution if 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 of the shifted log-logistic distribution is

for , where is the location parameter, the scale parameter and the shape parameter. Note that some references use to parameterise the shape.[3][4]

The probability density function is

again, for

The shape parameter is often restricted to lie in [-1,1], when the probability density function is bounded. When , it has an asymptote at . Reversing the sign of reflects the pdf and the cdf about .

Related distributions[edit]

  • When the shifted log-logistic reduces to the log-logistic distribution.
  • When → 0, the shifted log-logistic reduces to the logistic distribution.
  • The shifted log-logistic with shape parameter is the same as the generalized Pareto distribution with shape parameter


The three-parameter log-logistic distribution is used in hydrology for modelling flood frequency.[3][4][5]


  1. ^ Venter, Gary G. (Spring 1994), "Introduction to selected papers from the variability in reserves prize program" (PDF), Casualty Actuarial Society Forum 1: 91–101 
  2. ^ Geskus, Ronald B. (2001), "Methods for estimating the AIDS incubation time distribution when date of seroconversion is censored", Statistics in Medicine 20 (5): 795–812, doi:10.1002/sim.700, PMID 11241577 
  3. ^ a b c Hosking, Jonathan R. M.; Wallis, James R (1997), Regional Frequency Analysis: An Approach Based on L-Moments, Cambridge University Press, ISBN 0-521-43045-3 
  4. ^ a b Robson, A.; Reed, D. (1999), Flood Estimation Handbook, 3: "Statistical Procedures for Flood Frequency Estimation", Wallingford, UK: Institute of Hydrology, ISBN 0-948540-89-3 
  5. ^ Ahmad, M. I.; Sinclair, C. D.; Werritty, A. (1988), "Log-logistic flood frequency analysis", Journal of Hydrology 98 (3–4): 205–224, doi:10.1016/0022-1694(88)90015-7