The Generalized Gaussian density has the following form:

where (rho) is the "shape parameter". The density isplotted in the following figure:

Matlab code used to generate this figure is available here: ggplot.m.

Adding an arbitrary location parameter, , and inverse scaleparameter, , the density has the form,

Matlab code used to generate this figure is available here: ggplot2.m.

Samples from the Generalized Gaussian can be generated by a transformation of Gamma random samples, using the fact that if is a distributed random variable, and is an independent random variable taking the value -1 or +1with equal probability, then,

is distributed . That is,

where the density of is written in a non-standard butsuggestive form.

Matlab code to generate random variates from the Generalized Gaussian density with parameters as described here is here:

gg6.m

As an example, we generate random samples from the example Generalized Gaussian densities shown above.

Matlab code used to generate this figure is available here: ggplot3.m.

A more general family of densities can be constructed from mixtures of Generalized Gaussians. A mixture density, , is made up of constituent densities together with probabilities associated with each constituent density.

The densities have different forms, or parameter values. Arandom variable with a mixture density can be thought of as being generated by a two-part process: first a decision is made as to which constituent density to draw from, where the densityis chosen with probability , then the value of therandom variable is drawn from the chosen density. Independent repetitions of this process result in a sample having the mixture density . As an example consider the density,

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Matlab code used to generate these figures is available here: ggplot4.m.