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. Type I has also been called the skew-logistic distribution. Type IV subsumes the other types and is obtained when applying the logit transform to beta random variates. Following the same convention as for the log-normal distribution, type IV may be referred to as the logistic-beta distribution, with reference to the standard logistic function, which is the inverse of the logit transform. 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; and the metalog ("meta-logistic") distribution, which is highly shape-and-bounds flexible and can be fit to data with linear least squares. 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 (-∞,∞). The corresponding probability density function is: This type has also been called the "skew-logistic" distribution. The corresponding probability density function is: Here B is the beta function. The moment generating function for this type is The corresponding cumulative distribution function is: Where, B is the beta function and is the standard logistic function. The moment generating function for this type is This type is also called the "exponential generalized beta of the second type". The corresponding cumulative distribution function is: Type IV is the most general form of the distribution. The Type III distribution can be obtained from Type IV by fixing . The Type II distribution can be obtained from Type IV by fixing (and renaming to ). The Type I distribution can be obtained from Type IV by fixing . Fixing gives the standard logistic distribution.