Probability density function 

Cumulative distribution function 

parameters:  α > 0 scale β > 0 shape 

support:  
pdf:  
cdf:  
mean:  if β > 1, else undefined 
median:  
mode:  if β > 1, 0 otherwise 
variance:  See main text 
skewness:  
kurtosis:  
entropy:  
mgf:  
cf: 
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 
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 α > 0 is a scale parameter and is also the median of the distribution. The parameter β > 0 is a shape parameter. The distribution is unimodal when β > 1 and its dispersion decreases as β increases.
The cumulative distribution function is
where x > 0, α > 0, β > 0.
The probability density function is
The kth raw moment exists only when k < β, 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 b = π / β for convenience, the mean is
and the variance is
Explicit expressions for the skewness and kurtosis are lengthy.^{[5]} As β tends to infinity the mean tends to α, the variance and skewness tend to zero and the excess kurtosis tends to 6/5 (see also related distributions below).
The quantile function (inverse cumulative distribution function) is :
It follows that the median is α, the lower quartile is 3^{1 / β}α and the upper quartile is 3 ^{− 1 / β}α.
The loglogistic distribution provides one parametric model for survival analysis. Unlike the more commonlyused Weibull distribution, it can have a nonmonotonic hazard function: when β > 1, the hazard function is unimodal (when β ≤ 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 β to differ between groups, or more generally by introducing covariates that affect β but not α by modelling log(β) as a linear function of the covariates.^{[7]}
The survival function is
and so the hazard function is
The loglogistic distribution has been used in hydrology for modelling stream flow rates and precipitation.^{[1]}^{[2]}
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 1 / β.^{[9]}
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.
Probability density function 

Cumulative distribution function 

parameters: 
location (real) 

support: 

pdf:  where 
cdf:  where 
mean:  where 
median:  
mode:  
variance:  where 
skewness:  
kurtosis:  
entropy:  
mgf:  
cf: 
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 δ: if X has a loglogistic distribution then X + δ has a shifted loglogistic distribution. So Y has a shifted loglogistic distribution if log(Y − δ) 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 ξ is often restricted to lie in [1,1], when the probability density function is bounded. When  ξ  > 1, it has an asymptote at x = μ − σ / ξ. Reversing the sign of ξ reflects the pdf and the cdf about x = 0..
The threeparameter loglogistic distribution is used in hydrology for modelling flood frequency.^{[10]}^{[13]}^{[14]}
