Loglogistic distribution
Probability density function  
Cumulative distribution function  
Parameters  scale shape 

Support  
Probability density function (pdf)  
Cumulative distribution function (cdf)  
Mean  if , else undefined 
Median  
Mode  if , 0 otherwise 
Variance  See main text 
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
In probability and statistics, the loglogistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a nonnegative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, and in economics as a simple model of the distribution of wealth or income.
The loglogistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution. It is similar in shape to the lognormal distribution but has heavier tails. Its cumulative distribution function can be written in closed form, unlike that of the lognormal.
Contents
Characterisation
There are several different parameterizations of the distribution in use. The one shown here gives reasonably interpretable parameters and a simple form for the cumulative distribution function.^{[1]}^{[2]} The parameter is a scale parameter and is also the median of the distribution. The parameter is a shape parameter. The distribution is unimodal when and its dispersion decreases as increases.
The cumulative distribution function is
where , ,
The probability density function is
Properties
Moments
The th raw moment exists only when when it is given by^{[3]}^{[4]}
where B() is the beta function. Expressions for the mean, variance, skewness and kurtosis can be derived from this. Writing for convenience, the mean is
and the variance is
Explicit expressions for the kurtosis and variance are lengthy.^{[5]} As tends to infinity the mean tends to , the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below).
Quantiles
The quantile function (inverse cumulative distribution function) is :
It follows that the median is , the lower quartile is and the upper quartile is .
Applications
Survival analysis
The loglogistic distribution provides one parametric model for survival analysis. Unlike the more commonlyused Weibull distribution, it can have a nonmonotonic hazard function: when the hazard function is unimodal (when ≤ 1, the hazard decreases monotonically). The fact that the cumulative distribution function can be written in closed form is particularly useful for analysis of survival data with censoring.^{[6]} The loglogistic distribution can be used as the basis of an accelerated failure time model by allowing to differ between groups, or more generally by introducing covariates that affect but not by modelling as a linear function of the covariates.^{[7]}
The survival function is
and so the hazard function is
Hydrology
The loglogistic distribution has been used in hydrology for modelling stream flow rates and precipitation.^{[1]}^{[2]}
Economics
The loglogistic has been used as a simple model of the distribution of wealth or income in economics, where it is known as the Fisk distribution.^{[8]} Its Gini coefficient is .^{[9]}
Related distributions
 If X has a loglogistic distribution with scale parameter and shape parameter then Y = log(X) has a logistic distribution with location parameter and scale parameter .
 As the shape parameter of the loglogistic distribution increases, its shape increasingly resembles that of a (very narrow) logistic distribution. Informally, as →∞,
 The loglogistic distribution with shape parameter and scale parameter is the same as the generalized Pareto distribution with location parameter , shape parameter and scale parameter
Generalizations
Several different distributions are sometimes referred to as the generalized loglogistic distribution, as they contain the loglogistic as a special case.^{[9]} These include the Burr Type XII distribution (also known as the SinghMaddala distribution) and the Dagum distribution, both of which include a second shape parameter. Both are in turn special cases of the even more general generalized beta distribution of the second kind. Another more straightforward generalization of the loglogistic is given in the next section.
Shifted loglogistic distribution
Probability density function  
Cumulative distribution function  
Parameters  location (real) 

Support 

Probability density function (pdf)  where 
Cumulative distribution function (cdf)  where 
Mean  where 
Median  
Mode  
Variance  where 
Skewness  
Excess kurtosis  
Entropy  
Momentgenerating function (mgf)  
Characteristic function 
The shifted loglogistic distribution is also known as the generalized loglogistic, the generalized logistic, or the threeparameter loglogistic distribution.^{[10]} ^{[11]}^{[12]} It can be obtained from the loglogistic distribution by addition of a shift parameter : 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 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.^{[10]}^{[13]}
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
 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.^{[10]}^{[13]}^{[14]}
See also
References
 ↑ ^{1.0} ^{1.1} Shoukri, M.M.; Mian, I.U.M.; Tracy, D.S. (1988), "Sampling Properties of Estimators of the LogLogistic Distribution with Application to Canadian Precipitation Data", The Canadian Journal of Statistics, 16 (3): 223–236
 ↑ ^{2.0} ^{2.1} Ashkar, Fahim; Mahdi, Smail (2006), "Fitting the loglogistic distribution by generalized moments", Journal of Hydrology, 328: 694–703, doi:10.1016/j.jhydrol.2006.01.014
 ↑ Tadikamalla, Pandu R.; Johnson, Norman L. (1982), "Systems of Frequency Curves Generated by Transformations of Logistic Variables", Biometrika, 69 (2): 461–465
 ↑ Tadikamalla, Pandu R. (1980), "A Look at the Burr and Related Distributions", International Statistical Review, 48 (3): 337–344
 ↑ McLaughlin, Michael P. (2001), A Compendium of Common Probability Distributions (PDF), p. A37, retrieved 20080215
 ↑ Bennett, Steve (1983), "LogLogistic Regression Models for Survival Data", Applied Statistics, 32 (2): 165–171
 ↑ Collett, Dave (2003), Modelling Survival Data in Medical Research (2nd ed.), CRC press, ISBN 1584883251
 ↑ Fisk, P.R. (1961), "The Graduation of Income Distributions", Econometrica, 29: 171–185
 ↑ ^{9.0} ^{9.1} Kleiber, C.; Kotz, S (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley, ISBN 0471150649
 ↑ ^{10.0} ^{10.1} ^{10.2} Hosking, Jonathan R. M.; Wallis, James R (1997), Regional Frequency Analysis: An Approach Based on LMoments, Cambridge University Press, ISBN 0521430453
 ↑ 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
 ↑ ^{13.0} ^{13.1} 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: 205–224, doi:10.1016/00221694(88)900157