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. 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 or the probability density function, 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".
The corresponding cumulative distribution function is:
