Template:Loglogistic distribution characteristics

Distribution Characteristics
For $$\sigma >1$$ :


 * $$f(t)$$ decreases monotonically and is convex. Mode and mean do not exist.

For $$\sigma =1$$ :


 * $$f(t)$$ decreases monotonically and is convex. Mode and mean do not exist. As  $$t\to 0$$,  $$f(t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.$$
 * As $$t\to 0$$ ,  $$\lambda (t)\to \tfrac{1}{\sigma {{e}^{\tfrac{\mu }{\sigma }}}}.$$

For $$0<\sigma <1$$ :


 * The shape of the loglogistic distribution is very similar to that of the lognormal distribution and the Weibull distribution.
 * The $$pdf$$  starts at zero, increases to its mode, and decreases thereafter.
 * As $$\mu $$  increases, while  $$\sigma $$  is kept the same, the  $$pdf$$  gets stretched out to the right and its height decreases, while maintaining its shape.
 * As $$\mu $$  decreases,while  $$\sigma $$  is kept the same, the  ..  gets pushed in towards the left and its height increases.
 * $$\lambda (t)$$ increases till  $$t={{e}^{\mu +\sigma \ln (\tfrac{1-\sigma }{\sigma })}}$$   and decreases thereafter.  $$\lambda (t)$$  is concave at first, then becomes convex.