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

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

The corresponding probability density function is:

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

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

Type II

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

The corresponding probability density function is:

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

Type III

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

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

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

The corresponding cumulative distribution function is:

$F(x;\alpha)= \frac{\left(e^x+1\right) \Gamma (\alpha ) e^{\alpha (-x)} \left(e^{-x}+1\right)^{-2 \alpha } \, _2\tilde{F}_1\left(1,1-\alpha ;\alpha +1;-e^x\right)}{B(\alpha ,\alpha )}, \quad \alpha > 0 .$

Type IV

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

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

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

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

The corresponding cumulative distribution function is:

$F(x;\alpha,\beta)= \frac{\left(e^x+1\right) \Gamma (\alpha ) e^{\beta (-x)} \left(e^{-x}+1\right)^{-\alpha -\beta } \, _2\tilde{F}_1\left(1,1-\beta ;\alpha +1;-e^x\right)}{B(\alpha ,\beta )} , \quad \alpha,\beta > 0 .$

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)