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.

ParameterDescriptionSupport
muLog meanμ>0
sigmaLog scale parameterσ>0

Probability Density Function

The probability density function (pdf) is

f(x|μ,σ)=1σ1xez(1+ez)2;x0,

where z=log(x)μσ.

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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