Generalized Gaussian distribution

Generalized Gaussian Distribution (GGD)
A random variable X has generalized Gaussian distribution if its probability density function (pdf) is given by

$$f(x;m,\sigma,\alpha)=a\ \exp(-|(x-m)/b|^\alpha)$$ ,$$x \in \R $$

Where m is the mean of the distribution, $$\sigma$$ is the standard deviation, $$\alpha$$ is the shape parameter and $$\sigma, \alpha>0$$. a & b are computed according to :

$$a = \frac{1}{2\Gamma(1+1/\alpha)b }$$ $$b = \sigma\,\sqrt{\frac{\Gamma(1/\alpha)}{\Gamma(3/\alpha)}}$$

b is a scaling factor which allows the variance to be $$\sigma^2$$.

When $$\alpha=1$$, $$f(x;m,\sigma,\alpha)$$ corresponds to a Laplacian or double exponential distribution, $$\alpha=2$$ corresponds to a Gaussian distribution, whereas in the limiting cases where $$\alpha$$ approches $$+\infty$$ the pdf ( $$f(x;m,\sigma,\alpha)$$ ) converges to a uniform distribution in $$(m-\sqrt{3}\sigma, m+\sqrt{3}\sigma)$$.

As GGD is symmetric around its mean (m), even central moments are zero.

Reference:

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