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,  >0, \\ {t}'= & ln(t) \end{align}$$ where,
 * $$\begin{align}

\mu= & \text{scale parameter} \\ \sigma=& \text{shape parameter} \end{align}$$ The loglogistic distribution and its characteristics are presented in more detail in Chapter 15.