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The term '''generalized logistic distribution''' is used as the name for several different families of [[probability distributions]]. For example, Johnson et al.<ref name=J1>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) ''Continuous Univariate Distributions, Volume 2'', Wiley. ISBN 0-471-58494-0 (pages 140–142)</ref> 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 [[Log-logistic distribution#Shifted log-logistic distribution|shifted log-logistic distribution]], which is a generalization of the [[log-logistic distribution]].


==Definitions==


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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]] (''&fnof;''), and is defined on (-∞,∞).
 
===Type I===
:<math>F(x;\alpha)=\frac{1}{(1+\exp(-x))^\alpha} \equiv (1+\exp(-x))^{-\alpha}, \quad \alpha > 0 .</math>
The corresponding probability density function is:
:<math>f(x;\alpha)=\frac{\alpha \exp(-x)}{\left(1+\exp(-x)\right)^{\alpha+1}}, \quad \alpha > 0 .</math>
This type has also been called the "skew-logistic" distribution.
 
===Type II ===
:<math>F(x;\alpha)=1-\frac{\exp(-\alpha x)}{(1+\exp(-x))^\alpha}, \quad \alpha > 0 .</math>
 
===Type III ===
:<math>f(x;\alpha)=\frac{1}{B(\alpha,\alpha)}\frac{\exp(-\alpha x)}{(1+\exp(-x))^{2\alpha}}, \quad \alpha > 0 .</math>
Here ''B'' is the [[beta function]]. The [[moment generating function]] for this type is
:<math>M(t)=\frac{\Gamma(\alpha-t) \Gamma(\alpha+t) }{ (\Gamma(\alpha))^2 }, \quad -\alpha<t<\alpha.</math>  
 
===Type IV ===
:<math>f(x;\alpha,\beta)=\frac{1}{B(\alpha,\beta)}\frac{\exp(-\beta x)}{(1+\exp(-x))^{\alpha+\beta}}, \quad \alpha,\beta > 0 .</math>
Again, ''B'' is the [[beta function]]. The [[moment generating function]] for this type is
:<math>M(t)=\frac{\Gamma(\beta-t) \Gamma(\alpha+t) }{ \Gamma(\alpha) \Gamma(\beta) }, \quad -\alpha<t<\beta.</math>
This type is also called the "exponential generalized beta of the second type".<ref name=J1/>
 
==See also==
*[[Champernowne distribution]], another generalization of the logistic distribution.
 
==References==
<references/>
 
{{ProbDistributions|continuous-infinite}}
 
[[Category:Continuous distributions]]
[[Category:Probability distributions]]
{{statistics-stub}}

Latest revision as of 15:02, 17 July 2013

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;α)=1(1+exp⁡(−x))α≡(1+exp⁡(−x))−α,α>0.

The corresponding probability density function is:

f(x;α)=αexp⁡(−x)(1+exp⁡(−x))α+1,α>0.

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

Type II

F(x;α)=1−exp⁡(−αx)(1+exp⁡(−x))α,α>0.

Type III

f(x;α)=1B(α,α)exp⁡(−αx)(1+exp⁡(−x))2α,α>0.

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

M(t)=Γ(α−t)Γ(α+t)(Γ(α))2,−α<t<α.

Type IV

f(x;α,β)=1B(α,β)exp⁡(−βx)(1+exp⁡(−x))α+β,α,β>0.

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

M(t)=Γ(β−t)Γ(α+t)Γ(α)Γ(β),−α<t<β.

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

See also

References

  1. ↑ 1.0 1.1 Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0-471-58494-0 (pages 140–142)

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