Generalized logistic distribution

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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.[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[edit]

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[edit]

The corresponding probability density function is:

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

Type II[edit]

The corresponding probability density function is:

Type III[edit]

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

The corresponding cumulative distribution function is:

Type IV[edit]

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

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

The corresponding cumulative distribution function is:

See also[edit]

References[edit]

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