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. 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".

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.