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	金宝app
`mu`	Mean of logarithmic values	$μ > 0$
`sigma`	Scale parameter of logarithmic values	$σ > 0$

Probability Density Function

The probability density function (pdf) is

$f (x | μ, σ) = \frac{1}{σ} \frac{1}{x} \frac{e^{z}}{{(1 + e^{z})}^{2}}; x \geq 0,$

where $z = \frac{\log (x) - μ}{σ}$ .

Relationship to Other Distributions

The loglogistic distribution is closely related to the logistic distribution. Ifxis distributed loglogistically with parametersμandσ, then log(x) is distributed logistically with parametersμandσ. The relationship is similar to that between the对数正态andnormaldistribution.