Log-logistic Distribution - Characterisation

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. 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

$\begin{align} F(x; \alpha, \beta) & = { 1 \over 1+(x/\alpha)^{-\beta} } \\ & = {(x/\alpha)^\beta \over 1+(x/\alpha)^ \beta } \\ & = {x^\beta \over \alpha^\beta+x^\beta} \end{align}$

where,

The probability density function is

$f(x; \alpha, \beta) = \frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} } { \left^2 }.$

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