Shifted loglogistic distribution
Probability density function values of as shown in legend  
Cumulative distribution function values of as shown in legend  
Parameters  shape (real) 

Support 

 
CDF 

Mean 

Median  
Mode  
Variance 

The shifted loglogistic distribution is a probability distribution also known as the generalized loglogistic or the threeparameter loglogistic 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.
Definition
The shifted loglogistic distribution can be obtained from the loglogistic distribution by addition of a shift parameter . Thus if has a loglogistic distribution then has a shifted loglogistic distribution. So has a shifted loglogistic distribution if has a logistic distribution. The shift parameter adds a location parameter to the scale and shape parameters of the (unshifted) loglogistic.
The properties of this distribution are straightforward to derive from those of the loglogistic 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 loglogistic 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 (PDF) 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
 When the shifted loglogistic reduces to the loglogistic distribution.
 When → 0, the shifted loglogistic reduces to the logistic distribution.
 The shifted loglogistic with shape parameter is the same as the generalized Pareto distribution with shape parameter
Applications
The threeparameter loglogistic distribution is used in hydrology for modelling flood frequency.^{[3]}^{[4]}^{[5]}
Alternate parameterization
An alternate parameterization with simpler expressions for the PDF and CDF is as follows. For the shape parameter , scale parameter and location parameter , the PDF is given by ^{[6]}^{[7]}
The CDF is given by
The mean is and the variance is , where .^{[7]}
References
 ↑ Venter, Gary G. (Spring 1994), "Introduction to selected papers from the variability in reserves prize program" (PDF), Casualty Actuarial Society Forum, 1: 91–101
 ↑ 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
 1 2 3 Hosking, Jonathan R. M.; Wallis, James R (1997), Regional Frequency Analysis: An Approach Based on LMoments, Cambridge University Press, ISBN 0521430453
 1 2 Robson, A.; Reed, D. (1999), Flood Estimation Handbook, 3: "Statistical Procedures for Flood Frequency Estimation", Wallingford, UK: Institute of Hydrology, ISBN 0948540893
 ↑ Ahmad, M. I.; Sinclair, C. D.; Werritty, A. (1988), "Loglogistic flood frequency analysis", Journal of Hydrology, 98 (3–4): 205–224, doi:10.1016/00221694(88)900157
 ↑ "EasyFit  LogLogistic Distribution". Retrieved 1 October 2016.
 1 2 "Guide to Using  RISK7_EN.pdf" (PDF). Retrieved 1 October 2016.