Maximum likelihood estimator

It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method.^[5][6] With ${\displaystyle \mu }$ initially set to the sample first moment ${\displaystyle m_{1}}$, ${\displaystyle \textstyle \beta }$ is estimated by using a Newton–Raphson iterative procedure, starting from an initial guess of ${\displaystyle \textstyle \beta =\textstyle \beta _{0}}$,

${\displaystyle \beta _{0}={\frac {m_{1}}{\sqrt {m_{2}}}},}$

where

${\displaystyle m_{1}={1 \over N}\sum _{i=1}^{N}|x_{i}|,}$

is the first statistical moment of the absolute values and ${\displaystyle m_{2}}$ is the second statistical moment. The iteration is

${\displaystyle \beta _{i+1}=\beta _{i}-{\frac {g(\beta _{i})}{g'(\beta _{i})}},}$

where

${\displaystyle g(\beta )=1+{\frac {\psi (1/\beta )}{\beta }}-{\frac {\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\log |x_{i}-\mu |}{\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }}}+{\frac {\log({\frac {\beta }{N}}\sum _{i=1}^{N}|x_{i}-\mu |^{\beta })}{\beta }},}$

and

${\displaystyle {\begin{aligned}g'(\beta )={}&-{\frac {\psi (1/\beta )}{\beta ^{2}}}-{\frac {\psi '(1/\beta )}{\beta ^{3}}}+{\frac {1}{\beta ^{2}}}-{\frac {\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }(\log |x_{i}-\mu |)^{2}}{\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }}}\\[6pt]&{}+{\frac {\left(\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\log |x_{i}-\mu |\right)^{2}}{\left(\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\right)^{2}}}+{\frac {\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\log |x_{i}-\mu |}{\beta \sum _{i=1}^{N}|x_{i}-\mu |^{\beta }}}\\[6pt]&{}-{\frac {\log \left({\frac {\beta }{N}}\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\right)}{\beta ^{2}}},\end{aligned}}}$

and where ${\displaystyle \psi }$ and ${\displaystyle \psi '}$ are the digamma function and trigamma function.

Given a value for ${\displaystyle \textstyle \beta }$, it is possible to estimate ${\displaystyle \mu }$ by finding the minimum of:

${\displaystyle \min _{\mu }=\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }}$

Finally ${\displaystyle \textstyle \alpha }$ is evaluated as

${\displaystyle \alpha =\left({\frac {\beta }{N}}\sum _{i=1}^{N}|x_{i}-\mu |^{\beta }\right)^{1/\beta }.}$

For ${\displaystyle \beta \leq 1}$, median is a more appropriate estimator of ${\displaystyle \mu }$ . Once ${\displaystyle \mu }$ is estimated, ${\displaystyle \beta }$ and ${\displaystyle \alpha }$ can be estimated as described above. ^[7]

Moments

Let ${\displaystyle X_{\beta }}$ be zero mean generalized Gaussian distribution of shape ${\displaystyle \beta }$ and scaling parameter ${\displaystyle \alpha }$ . The moments of ${\displaystyle X_{\beta }}$ exist and are finite for any k greater than −1. For any non-negative integer k, the plain central moments are^[2]

${\displaystyle \operatorname {E} \left[X_{\beta }^{k}\right]={\begin{cases}0&{\text{if }}k{\text{ is odd,}}\\\alpha ^{k}\Gamma \left({\frac {k+1}{\beta }}\right){\Big /}\,\Gamma \left({\frac {1}{\beta }}\right)&{\text{if }}k{\text{ is even.}}\end{cases}}}$

Connection to Stable Count Distribution

From the viewpoint of the Stable count distribution, ${\displaystyle \beta }$ can be regarded as Lévy's stability parameter. This distribution can be decomposed to an integral of kernel density where the kernel is either a Laplace distribution or a Gaussian distribution:

${\displaystyle {\frac {1}{2}}{\frac {1}{\Gamma ({\frac {1}{\beta }}+1)}}e^{-z^{\beta }}={\begin{cases}\displaystyle \int _{0}^{\infty }{\frac {1}{\nu }}\left({\frac {1}{2}}e^{-|z|/\nu }\right){\mathfrak {N}}_{\beta }(\nu )\,d\nu ,&1\geq \beta >0;{\text{or }}\\\displaystyle \int _{0}^{\infty }{\frac {1}{s}}\left({\frac {1}{\sqrt {2\pi }}}e^{-{\frac {1}{2}}(z/s)^{2}}\right)V_{\beta }(s)\,ds,&2\geq \beta >0;\end{cases}}}$

where ${\displaystyle {\mathfrak {N}}_{\beta }(\nu )}$ is the Stable count distribution and ${\displaystyle V_{\beta }(s)}$ is the Stable vol distribution.

Connection to Positive-Definite Functions

The probability density function of the symmetric generalized normal distribution is a positive-definite function for ${\displaystyle \beta \in (0,2]}$.^[10][11]

Infinite divisibility

The symmetric generalized Gaussian distribution is an infinitely divisible distribution if and only if ${\displaystyle \beta \in (0,1]\cup \{2\}}$.^[12]