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Concept# Log-logistic distribution

Summary

In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative 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.
The log-logistic distribution is the probability distribution of a random variable whose logarithm has a logistic distribution.
It is similar in shape to the log-normal distribution but has heavier tails. Unlike the log-normal, 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.
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
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.
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 .

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