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, as, for example, mortality rate from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.
Probability density function  
Cumulative distribution function  
Parameters 
scale shape  

Support  
CDF  
Quantile  
Mean 
if , else undefined  
Median  
Mode 
if , 0 otherwise  
Variance  See main text  
Entropy  
MGF  ^{[1]} where is the Beta function.^{[2]}  
CF  ^{[1]} where is the Beta function.^{[2]}  
Expected shortfall 
where is the incomplete beta function.^{[3]} 
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. Unlike the lognormal, its cumulative distribution function can be written in closed form.
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.^{[4]}^{[5]} 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
An alternative parametrization is given by the pair in analogy with the logistic distribution:
The th raw moment exists only when when it is given by^{[6]}^{[7]}
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 skewness and kurtosis are lengthy.^{[8]} 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).
The quantile function (inverse cumulative distribution function) is :
It follows that the median is , the lower quartile is and the upper quartile is .
The loglogistic distribution provides one parametric model for survival analysis. Unlike the more commonly used 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.^{[9]} 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.^{[10]}
The survival function is
and so the hazard function is
The loglogistic distribution with shape parameter is the marginal distribution of the intertimes in a geometricdistributed counting process.^{[11]}
The loglogistic distribution has been used in hydrology for modelling stream flow rates and precipitation.^{[4]}^{[5]}
Extreme values like maximum oneday rainfall and river discharge per month or per year often follow a lognormal distribution.^{[12]} The lognormal distribution, however, needs a numeric approximation. As the loglogistic distribution, which can be solved analytically, is similar to the lognormal distribution, it can be used instead.
The blue picture illustrates an example of fitting the loglogistic distribution to ranked maximum oneday October rainfalls and it shows the 90% confidence belt based on the binomial distribution. The rainfall data are represented by the plotting position r/(n+1) as part of the cumulative frequency analysis.
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.^{[13]} Its Gini coefficient is .^{[14]}
Derivation of Gini coefficient


The Gini coefficient for a continuous probability distribution takes the form: where is the CDF of the distribution and is the expected value. For the loglogistic distribution, the formula for the Gini coefficient becomes: Defining the substitution leads to the simpler equation: And making the substitution further simplifies the Gini coefficient formula to: The integral component is equivalent to the standard beta function . The beta function may also be written as: where is the gamma function. Using the properties of the gamma function, it can be shown that: From Euler's reflection formula, the expression can be simplified further: Finally, we may conclude that the Gini coefficient for the loglogistic distribution . 
The loglogistic has been used as a model for the period of time beginning when some data leaves a software user application in a computer and the response is received by the same application after travelling through and being processed by other computers, applications, and network segments, most or all of them without hard realtime guarantees (for example, when an application is displaying data coming from a remote sensor connected to the Internet). It has been shown to be a more accurate probabilistic model for that than the lognormal distribution or others, as long as abrupt changes of regime in the sequences of those times are properly detected.^{[15]}
Several different distributions are sometimes referred to as the generalized loglogistic distribution, as they contain the loglogistic as a special case.^{[14]} These include the Burr Type XII distribution (also known as the Singh–Maddala 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 the shifted loglogistic distribution.
Another generalized loglogistic distribution is the logtransform of the metalog distribution, in which power series expansions in terms of are substituted for logistic distribution parameters and . The resulting logmetalog distribution is highly shape flexible, has simple closed form PDF and quantile function, can be fit to data with linear least squares, and subsumes the loglogistic distribution is special case.
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