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.

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 the interval .

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

Where, B is the beta function 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:

Relationship between types

gives the standard logistic distribution.

Type IV (logistic-beta) properties

The Type IV generalized logistic, or logistic-beta, has (as shown above) the probability density function (pdf):

is the standard logistic function. The probability density functions for three different sets of shape parameters are shown in the plot, where the distributions have been scaled and shifted to give zero means and unity variances, in order to facilitate comparison of the shapes.

is used to denote the Type IV distribution.

Relationship with Beta Distribution

Relationship with Gamma Distribution

This distribution can be obtained in terms of the gamma distribution and independently.

Symmetry

.

Normal variance-mean mixture representation

Logistic-beta distribution admits the following normal variance-mean mixture representation:

.

Mean and variance

By using the logarithmic expectations of the gamma distribution, the mean and variance can be derived as:

is the digamma function is its first derivative, also known as the trigamma function, or the first polygamma function is strictly increasing are increased. The two terms of the variance represent the contributions to the variance of the left and right parts of the distribution.

Cumulants and skewness

The cumulant generating function is given above. The cumulants:

, is the variance.

, which when scaled by the third power of the standard deviation gives the skewness:

The sign (and therefore the handedness.

Mode

where the log pdf derivative is zero:

, so that:

Tail behaviour

. Although type IV tails are heavier than those of the normal distribution. This is in contrast with the Cauchy distribution for which the mean and variance do not exist. In the log pdf plots shown here, the type IV tails are linear, the normal distribution tails are quadratic and the Cauchy tails are logarithmic.

Exponential family properties

forms an exponential family with natural parameters and sufficient statistics. The expected values of the sufficient statistics can be found by differentiation of the log-normalizer:

assumed to have been generated IID, the maximum-likelihood parameter estimate is:

where the overlines denote the averages of the sufficient statistics. The maximum-likelihood estimate depends on the data only via these average statistics. Indeed, at the maximum-likelihood estimate the expected values and averages agree:

which is also where the partial derivatives of the above maximand vanish.

Relationships with other distributions

Relationships with other distributions include:

• The log-ratio of gamma variates is of type IV as detailed above.

• has a type IV. See beta prime distribution.

• has a compound gamma distribution has a type IV distribution.

• has a type IV. See beta distribution. The logit function is the inverse of the logistic function. This relationship explains the name logistic-beta for this distribution: if the logistic function is applied to logistic-beta variates, the transformed distribution is beta.

Large shape parameters

, the distribution becomes more Gaussian, with:

This is demonstrated in the pdf and log pdf plots here.

Random variate generation

Since random sampling from the gamma and beta distributions are readily available on many software platforms, the above relationships with those distributions can be used to generate variates from the type IV distribution.

Generalization with location and scale parameters

A flexible, four-parameter family can be obtained by adding location and scale parameters. Moreover maximum-likelihood estimation with this parametrization is hard. These problems can be addressed as follows.

are:

, via the transformation:

to be either positive or negative does not generalize this family, because of the above-noted symmetry for this family.

is:

. The pdf and log-pdf plots above, where the captions contain (means=0, variances=1),.

Maximum likelihood parameter estimation

In this section, maximum-likelihood.

Maximum likelihood for standard Type IV

As noted above is an exponential family, the maximum-likelihood estimates of which depend only on averaged sufficient statistics:

Once these statistics have been accumulated, the maximum-likelihood estimate is given by:

an unconstrained numerical optimization algorithm like BFGS can be used. Optimization iterations are fast, because they are independent of the size of the data-set.

An alternative is to use an EM-algorithm. Because of the self-conjugacy of the gamma distribution that are required for the E-step can be computed in closed form. The M-step parameter update can be solved analogously to maximum-likelihood for the gamma distribution.

Maximum likelihood for the four-parameter family

is:

This is no longer an exponential family, so that each optimization iteration has to traverse the whole data-set. Moreover the computation of the partial derivatives (as required for example by BFGS) is considerably more complex than for the above two-parameter case. However, all the component functions are readily available in software packages with automatic differentiation. Again, the positive parameters can be parametrized in terms of their logarithms to obtain an unconstrained numerical optimization problem.

can be set to the empirical mean and variance of the data.

See also

• Champernowne distribution, another generalization of the logistic distribution.

References