## Loglogistic Distribution

### Overview

The loglogistic distribution is a probability distribution whose
logarith 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.

### Parameters

The loglogistic distribution uses the following parameters.

Parameter | Description | Support |

`mu` | Log mean | $$\mu >0$$ |

`sigma` | Log scale parameter | $$\sigma >0$$ |

### Probability Density Function

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

### Relationship to Other Distributions

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 mean
and standard deviation. The relationship is similar to that between
the lognormal and normal distribution.

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