Talk:Logistic distribution/Generalized log-logistic distribution

Generalized log-logistic distribution

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

Generalized log-logistic

Parameters	$\mu \in (-\infty ,\infty )\,$ location (real) $\sigma \in (0,\infty )\,$ scale (real) $\xi \in (-\infty ,\infty )\,$ shape (real)
Support	$x\geqslant \mu -\sigma /\xi \,\;(\xi \geqslant 0)$ $x\leqslant \mu -\sigma /\xi \,\;(\xi <0)$
PDF	${\frac {(1+\xi z)^{-(1/\xi +1)}}{\sigma \left(1+(1+\xi z)^{-1/\xi }\right)^{2}}}$ where $z=(x-\mu )/\sigma \,$
CDF	$\left(1+(1+\xi z)^{-1/\xi }\right)^{-1}\,$ where $z=(x-\mu )/\sigma \,$
Mean	$\mu +{\frac {\sigma }{\xi }}(\alpha \csc(\alpha )-1)$ where $\alpha =\pi \xi \,$
Median	$\mu \,$
Mode	$\mu +{\frac {\sigma }{\xi }}\left[\left({\frac {1-\xi }{1+\xi }}\right)^{\xi }-1\right]$
Variance	${\frac {\sigma ^{2}}{\xi ^{2}}}[2\alpha \csc(2\alpha )-(\alpha \csc(\alpha ))^{2}]$ where $\alpha =\pi \xi \,$

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