Generalized normal distribution

Parameter estimation via maximum likelihood and the method of moments has been studied.[3] The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed.[4]

The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C^∞ of smooth functions) only if ${\displaystyle \textstyle \beta }$ is a positive, even integer. Otherwise, the function has ${\displaystyle \textstyle \lfloor \beta \rfloor }$ continuous derivatives. As a result, the standard results for consistency and asymptotic normality of maximum likelihood estimates of ${\displaystyle \beta }$ only apply when ${\displaystyle \textstyle \beta \geq 2}$.

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]

The symmetric generalized normal distribution has been used in modeling when the concentration of values around the mean and the tail behavior are of particular interest.[8][9] Other families of distributions can be used if the focus is on other deviations from normality. If the symmetry of the distribution is the main interest, the skew normal family or asymmetric version of the generalized normal family discussed below can be used. If the tail behavior is the main interest, the student t family can be used, which approximates the normal distribution as the degrees of freedom grows to infinity. The t distribution, unlike this generalized normal distribution, obtains heavier than normal tails without acquiring a cusp at the origin.

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]

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

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.

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

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

The multivariate generalized normal distribution, i.e. the product of ${\displaystyle n}$ exponential power distributions with the same ${\displaystyle \beta }$ and ${\displaystyle \alpha }$ parameters, is the only probability density that can be written in the form ${\displaystyle p(\mathbf {x} )=g(\|\mathbf {x} \|_{\beta })}$ and has independent marginals.[13] The results for the special case of the Multivariate normal distribution is originally attributed to Maxwell.[14]

Parameters can be estimated via maximum likelihood estimation or the method of moments. The parameter estimates do not have a closed form, so numerical calculations must be used to compute the estimates. Since the sample space (the set of real numbers where the density is non-zero) depends on the true value of the parameter, some standard results about the performance of parameter estimates will not automatically apply when working with this family.

The asymmetric generalized normal distribution can be used to model values that may be normally distributed, or that may be either right-skewed or left-skewed relative to the normal distribution. The skew normal distribution is another distribution that is useful for modeling deviations from normality due to skew. Other distributions used to model skewed data include the gamma, lognormal, and Weibull distributions, but these do not include the normal distributions as special cases.