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
The corresponding probability density function is:
This type has also been called the "skew-logistic" distribution.
Type II
The corresponding probability density function is:
Type III
Here B is the beta function. The moment generating function for this type is
The corresponding cumulative distribution function is:
Type IV
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
- Champernowne distribution, another generalization of the logistic distribution.
References
- 1 2 Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley. ISBN 0-471-58494-0 (pages 140–142)