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" and "b" are computed according to :
:$a\; =\; frac\{1\}\{2Gamma(1+1/alpha)b\; \},\; quadb\; =\; sigma,sqrt\{frac\{Gamma(1/alpha)\}\{Gamma(3/alpha).$ "b" is a scaling factor which allows the variance to be $sigma^2$.$Gamma(cdot)$ stands for the Gamma function.

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

As GGD is symmetric about its mean (m), odd-order central moments are zero. [ J. Armando Domínguez-Molina, Graciela González-Farías,Ramón M. Rodríguez-Dagnino," [http://www.cimat.mx/reportes/enlinea/I-01-18_eng.pdf A practical procedure to estimate the shape parameter in the generalized Gaussian distribution] ".]

References

2.Varanasi, M.K., Aazhang, B. (1989). Parametric generalized Gaussian density estimation,J. Acoust. Soc. Am. 86 (4), October 1989, pp. 1404.