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:
 * $$ \begin{align}

f(t)= & \frac{e^z}{\sigma t{(1+{e^z})^2}} \\ z= & \frac{T'-{\mu }'}{\sigma } \\ f(t)\ge & 0,t>0,{{\sigma }_{T'}}>0, \\ {T}'= & ln(t) \end{align}$$
 * where,


 * $$\begin{align}

\mu'= & \text{scale parameter} \\ \sigma_{T}=& \text{shape parameter} \end{align}$$ The loglogistic distribution and its characteristics are presented in more detail in Chapter 10.