Generalized logistic distribution

The term generalized logistic distribution is used as the name for several different families of probability distributions. For example, Johnson et al.[1] list four forms, which are listed below. One family described here has also been called the skew-logistic distribution. For other families of distributions that have also been called generalized logistic distributions, see the shifted log-logistic distribution, which is a generalization of the log-logistic distribution.

Definitions

The following definitions are for standardized versions of the families, which can be expanded to the full form as a location-scale family. Each is defined using either the cumulative distribution function (F) or the probability density function (ƒ), and is defined on (-∞,∞).

Type I

${\displaystyle F(x;\alpha )={\frac {1}{(1+e^{-x})^{\alpha }}}\equiv (1+e^{-x})^{-\alpha },\quad \alpha >0.}$

The corresponding probability density function is:

${\displaystyle f(x;\alpha )={\frac {\alpha e^{-x}}{\left(1+e^{-x}\right)^{\alpha +1}}},\quad \alpha >0.}$

This type has also been called the "skew-logistic" distribution.

Type II

${\displaystyle F(x;\alpha )=1-{\frac {e^{-\alpha x}}{(1+e^{-x})^{\alpha }}},\quad \alpha >0.}$

The corresponding probability density function is:

${\displaystyle f(x;\alpha )={\frac {\alpha e^{-\alpha x}}{(1+e^{-x})^{\alpha +1}}},\quad \alpha >0.}$

Type III

${\displaystyle f(x;\alpha )={\frac {1}{B(\alpha ,\alpha )}}{\frac {e^{-\alpha x}}{(1+e^{-x})^{2\alpha }}},\quad \alpha >0.}$

Here B is the beta function. The moment generating function for this type is

${\displaystyle M(t)={\frac {\Gamma (\alpha -t)\Gamma (\alpha +t)}{(\Gamma (\alpha ))^{2}}},\quad -\alpha <t<\alpha .}$

The corresponding cumulative distribution function is:

${\displaystyle F(x;\alpha )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\alpha (-x)}\left(e^{-x}+1\right)^{-2\alpha }\,_{2}{\tilde {F}}_{1}\left(1,1-\alpha$ ;\alpha +1;-e^{x}\right)}{B(\alpha ,\alpha )}},\quad \alpha >0.}

Type IV

${\displaystyle f(x;\alpha ,\beta )={\frac {1}{B(\alpha ,\beta )}}{\frac {e^{-\beta x}}{(1+e^{-x})^{\alpha +\beta }}},\quad \alpha ,\beta >0.}$

Again, B is the beta function. The moment generating function for this type is

${\displaystyle M(t)={\frac {\Gamma (\beta -t)\Gamma (\alpha +t)}{\Gamma (\alpha )\Gamma (\beta )}},\quad -\alpha <t<\beta .}$

This type is also called the "exponential generalized beta of the second type".[1]

The corresponding cumulative distribution function is:

${\displaystyle F(x;\alpha ,\beta )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\beta (-x)}\left(e^{-x}+1\right)^{-\alpha -\beta }\,_{2}{\tilde {F}}_{1}\left(1,1-\beta$ ;\alpha +1;-e^{x}\right)}{B(\alpha ,\beta )}},\quad \alpha ,\beta >0.}

Relationship

Type IV is the most general form of the distribution. The Type III distribution can be obtained from Type IV by fixing ${\displaystyle \beta =\alpha }$. The Type II distribution can be obtained from Type IV by fixing ${\displaystyle \alpha =1}$ (and renaming ${\displaystyle \beta }$ to ${\displaystyle \alpha }$). The Type I distribution can be obtained from Type IV by fixing ${\displaystyle \beta =1}$.

See also

• Champernowne distribution, another generalization of the logistic distribution.

References

1. Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0-471-58494-0 (pages 140–142)