Talk:Logistic distribution/Generalized log-logistic distribution

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[edit] Generalized log-logistic distribution

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

Generalized log-logistic
Probability density function
Cumulative distribution function
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)
Probability density function (pdf) \frac{(1+\xi z)^{-(1/\xi +1)}}{\sigma\left(1 + (1+\xi z)^{-1/\xi}\right)^2}

where z=(x-\mu)/\sigma\,
Cumulative distribution function (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\,

Skewness {{{skewness}}}
Excess kurtosis {{{kurtosis}}}
Entropy
Moment-generating function (mgf)
Characteristic function

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.