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.

Parameter	Description	Support
mu	Mean of logarithmic values	$$\mu >0$$
sigma	Scale parameter of logarithmic values	$$\sigma >0$$

Probability Density Function

The probability density function (pdf) is

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

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

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.