Template:LoglogisticDistribution

The Loglogistic Distribution

As may be summarized from the name, the loglogistic distribution is similar to the logistic distribution. Specifically, the data follows a loglogistic distribution when the natural logarithms of the times-to-failure follow a logistic distribution. Accordingly, the loglogistic and lognormal distributions also share many similarities.
The pdf of the loglogistic distribution is given by:

[math]\displaystyle{ \begin{align} f(t)= & \frac{e^z}{\sigma{t}{(1+{e^z})^2}} \\ z= & \frac{t'-{\mu }}{\sigma } \\ f(t)\ge & 0, t\gt 0, {{\sigma}}\gt 0, \\ {t}'= & ln(t) \end{align} }[/math]

where,

[math]\displaystyle{ \begin{align} \mu= & \text{scale parameter} \\ \sigma=& \text{shape parameter} \end{align} }[/math]

The loglogistic distribution and its characteristics are presented in detail in The Loglogistic.