## Loglogistic Distribution

### Overview

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.

### Parameters

The loglogistic distribution uses the following parameters.

ParameterDescriptionSupport
`mu`Mean of logarithmic values$\mu >0$
`sigma`Scale parameter of logarithmic values$\sigma >0$

### Probability Density Function

The probability density function (pdf) is

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

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 parameters μ and σ. The relationship is similar to that between the lognormal and normal distribution.