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The loglogistic distribution is a probability distribution whose logarithm has a logistic distribution. This distribution is often used in survival analysis to model events that experience an initial rate increase, followed by a rate decrease. It is also known as the Fisk distribution in economics applications.

The loglogistic distribution uses the following parameters.

Parameter | Description | Support |
---|---|---|

`mu` | Mean of logarithmic values | $$\mu >0$$ |

`sigma` | Scale parameter of logarithmic values | $$\sigma >0$$ |

The probability density function (pdf) is

$$f(x|\mu ,\sigma )=\frac{1}{\sigma}\frac{1}{x}\frac{{e}^{z}}{{\left(1+{e}^{z}\right)}^{2}}\text{\hspace{1em}};\text{\hspace{1em}}x\ge 0\text{\hspace{0.17em}},$$

where $$z=\frac{\mathrm{log}\left(x\right)-\mu}{\sigma}$$.

The loglogistic distribution is closely related to the logistic distribution. If
*x* is distributed loglogistically with parameters
*μ* and *σ*, then log(*x*) is
distributed logistically with parameters *μ* and
*σ*. The relationship is similar to that between the lognormal and normal
distribution.