Talk:Logistic distribution/Generalized log-logistic distribution

Generalized log-logistic distribution
The Generalized log-logistic distribution (GLL) has three parameters $$ \mu,\sigma \,$$ and $$ \xi$$.

The cumulative distribution function is
 * $$F_{(\xi,\mu,\sigma)}(x) = \left(1 + \left(1+ \frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right)^{-1}$$

for $$ 1 + \xi(x-\mu)/\sigma \geqslant 0$$, where $$\mu\in\mathbb R$$ is the location parameter, $$\sigma>0 \,$$ the scale parameter and $$\xi\in\mathbb R$$ the shape parameter. Note that some references give the "shape parameter" as $$ \kappa = - \xi \,$$.

The probability density function is


 * $$\frac{\left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-(1/\xi +1)}}

{\sigma\left[1 + \left(1+\frac{\xi(x-\mu)}{\sigma}\right)^{-1/\xi}\right]^2}. $$

again, for $$ 1 + \xi(x-\mu)/\sigma \geqslant 0. $$