The term generalized logistic distribution is used as the name for several different families of

probability distributions In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

. For example, Johnson et al.Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) ''Continuous Univariate Distributions, Volume 2'', Wiley. (pages 140–142) list four forms, which are listed below. The Type I family described below 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 The shifted log-logistic distribution is a probability distribution also known as the generalized log-logistic or the three-parameter log-logistic distribution. It has also been called the generalized logistic distribution, but this conflicts with ...

, which is a generalization of the

log-logistic distribution In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events ...

; and the metalog ("meta-logistic") distribution, which is highly shape-and-bounds flexible and can be fit to data with linear least squares.

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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...

(''F'') or the

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

(''ƒ''), and is defined on (-∞,∞).

Type I

F(x;\alpha)=\frac \equiv (1+e^)^, \quad \alpha > 0 .

The corresponding probability density function is: :

f(x;\alpha)=\frac, \quad \alpha > 0 .

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

Type II

F(x;\alpha)=1-\frac, \quad \alpha > 0 .

The corresponding probability density function is: :

f(x;\alpha)=\frac, \quad \alpha > 0 .

Type III

f(x;\alpha)=\frac\frac, \quad \alpha > 0 .

Here ''B'' is the

beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1 ...

. The

moment generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compare ...

for this type is :

M(t)=\frac, \quad -\alpha The corresponding cumulative distribution function is:
: F(x;\alpha)= \frac, \quad \alpha > 0 .

Type IV

f(x;\alpha,\beta)=\frac\frac, \quad \alpha,\beta > 0 .

Again, ''B'' is the

. The

for this type is :

M(t)=\frac, \quad -\alpha This type is also called the "exponential generalized beta of the second type". The corresponding cumulative distribution function is:
: F(x;\alpha,\beta)= \frac , \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

\beta = \alpha

. The Type II distribution can be obtained from Type IV by fixing

\alpha = 1

(and renaming

\beta

\alpha

). The Type I distribution can be obtained from Type IV by fixing

\beta = 1

References

Continuous distributions {{statistics-stub