Generalized Gaussian distribution

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[edit] 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, $σ$ is the standard deviation, $α$ is the shape parameter and $σ,α > 0$ . a and b are computed according to :

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

b is a scaling factor which allows the variance to be $σ 2$ . $\Gamma(\cdot)$ stands for the Gamma function.

When $α = 1$ , $f (x; m,σ,α)$ corresponds to a Laplacian or double exponential distribution, $α = 2$ corresponds to a Gaussian distribution, whereas in the limiting cases where $α$ approaches $+\infty$ the pdf ( $f (x; m,σ,α)$ ) converges to a uniform distribution in $(m-\sqrt{3}\sigma, m+\sqrt{3}\sigma)$ . When $\alpha \rightarrow 0^+$ a degenerate distribution in $x = μ$ is obtained.