Talk:Logistic distribution/Generalized log-logistic distribution

From Wikipedia, the free encyclopedia
Jump to: navigation, search

Generalized log-logistic distribution[edit]

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.