Thegeneralized normal distribution (GND) orgeneralized Gaussian distribution (GGD) is either of two families ofparametriccontinuous probability distributions on thereal line. Both families add ashape parameter to thenormal distribution. To distinguish the two families, they are referred to below as "symmetric" and "asymmetric"; however, this is not a standard nomenclature.

Symmetric Generalized Normal
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \mu }$location (real) ${\displaystyle \alpha }$scale (positive,real) ${\displaystyle \beta }$shape (positive,real)
Support	${\displaystyle x\in (-\mathrm{\infty };+\mathrm{\infty })}$
PDF	${\displaystyle \frac{\beta }{2\alpha \mathrm{\Gamma }(1/\beta )}{e}^{-(|x-\mu |/\alpha {)}^{\beta }}}$ ${\displaystyle \mathrm{\Gamma }}$ denotes thegamma function
CDF	${\displaystyle \frac{1}{2}+\text{sign}(x-\mu )\frac{1}{2\mathrm{\Gamma }(1/\beta )}\gamma \left(1/\beta ,{\left|\frac{x-\mu }{\alpha }\right|}^{\beta }\right)}$ where${\displaystyle \beta }$ is a shape parameter,${\displaystyle \alpha }$ is a scale parameter and${\displaystyle \gamma }$ is theunnormalized incomplete lower gamma function.
Quantile	${\displaystyle \text{sign}(p-0.5){\left[{\alpha }^{\beta }{F}^{-1}\left(2|p-0.5|;\frac{1}{\beta }\right)\right]}^{1/\beta }+\mu }$ where${\displaystyle F^{-1}\left(p;a\right)}$ is the quantile function ofGamma distribution[1]
Mean	${\displaystyle \mu }$
Median	${\displaystyle \mu }$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle \frac{{\alpha }^2\mathrm{\Gamma }(3/\beta )}{\mathrm{\Gamma }(1/\beta )}}$
Skewness	0
Excess kurtosis	${\displaystyle \frac{\mathrm{\Gamma }(5/\beta )\mathrm{\Gamma }(1/\beta )}{\mathrm{\Gamma }(3/\beta {)}^2}-3}$
Entropy	${\displaystyle \frac{1}{\beta }-\log \left[\frac{\beta }{2\alpha \mathrm{\Gamma }(1/\beta )}\right]}$[2]

Thesymmetric generalized normal distribution, also known as theexponential power distribution or thegeneralized error distribution, is a parametric family ofsymmetric distributions. It includes allnormal andLaplace distributions, and as limiting cases it includes allcontinuous uniform distributions on bounded intervals of the real line.

This family includes thenormal distribution when${\displaystyle \beta =2}$ (with mean${\displaystyle \mu }$ and variance${\displaystyle \frac{{\alpha }^2}{2}}$) and it includes theLaplace distribution when${\displaystyle \beta =1}$. As${\displaystyle \beta \to \mathrm{\infty }}$, the densityconverges pointwise to a uniform density on${\displaystyle (\mu -\alpha ,\mu +\alpha )}$.

This family allows for tails that are either heavier than normal (when) or lighter than normal (when${\displaystyle \beta >2}$). It is a useful way to parametrize a continuum of symmetric,platykurtic densities spanning from the normal (${\displaystyle \beta =2}$) to the uniform density (${\displaystyle \beta =\mathrm{\infty }}$), and a continuum of symmetric,leptokurtic densities spanning from the Laplace (${\displaystyle \beta =1}$) to the normal density (${\displaystyle \beta =2}$).The shape parameter${\displaystyle \beta }$ also controls thepeakedness in addition to the tails.

Parameter estimation viamaximum likelihood and themethod 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^∞ ofsmooth functions) only if${\displaystyle \beta }$ is a positive, even integer. Otherwise, the function has${\displaystyle \lfloor \beta \rfloor }$ continuous derivatives. As a result, the standard results for consistency and asymptotic normality ofmaximum likelihood estimates of${\displaystyle \beta }$ only apply when${\displaystyle \beta \ge 2}$.

It is possible to fit the generalized normal distribution adopting an approximatemaximum likelihood method.^[5][6] With${\displaystyle \mu }$ initially set to the sample first moment${\displaystyle m_1}$,${\displaystyle \beta }$ is estimated by using aNewton–Raphson iterative procedure, starting from an initial guess of${\displaystyle \beta ={\beta }_0}$,

${\displaystyle {\beta }_0=\frac{m_1}{\sqrt{m_2}},}$

where

${\displaystyle m_1=\frac{1}{N}\sum _{i=1}^N|x_i|,}$

is the first statisticalmoment of the absolute values and${\displaystyle m_2}$ is the second statisticalmoment. The iteration is

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

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

and where${\displaystyle \psi }$ and${\displaystyle {\psi }^{\prime }}$ are thedigamma function andtrigamma function.

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

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

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

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

For${\displaystyle \beta \le 1}$, 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 thesymmetry of the distribution is the main interest, theskew normal family or asymmetric version of the generalized normal family discussed below can be used. If the tail behavior is the main interest, thestudent 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 acusp at the origin. It finds uses in plasma physics under the name of Langdon Distribution resulting from inverse bremsstrahlung.^[10]

In alinear regression problem modeled as${\displaystyle y\sim \mathrm{G}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{l}(X\cdot \theta ,\alpha ,p)}$, theMLE will be the${\displaystyle \arg \underset{\theta }{min}\Vert X\cdot \theta -y{\Vert }_p}$ where thep-norm is used.

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]

From the viewpoint of theStable 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 aLaplace distribution or aGaussian distribution:

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

The probability density function of the symmetric generalized normal distribution is apositive-definite function for${\displaystyle \beta \in (0,2]}$.^[11][12]

The symmetric generalized Gaussian distribution is aninfinitely divisible distribution if and only if${\displaystyle \beta \in (0,1]\cup \{2\}}$.^[11]

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${\displaystyle p(\mathbf{x})=g(\Vert \mathbf{x}{\Vert }_{\beta })}$ and has independent marginals.^[13] The results for the special case of theMultivariate normal distribution is originally attributed toMaxwell.^[14]

Asymmetric Generalized Normal
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \xi }$location (real) ${\displaystyle \alpha }$scale (positive,real) ${\displaystyle \kappa }$shape (real)
Support	${\displaystyle x\in (-\mathrm{\infty },\xi +\alpha /\kappa )\text{ if }\kappa >0}$ ${\displaystyle x\in (-\mathrm{\infty },\mathrm{\infty })\text{ if }\kappa =0}$
PDF	${\displaystyle \frac{\varphi (y)}{\alpha -\kappa (x-\xi )}}$, where ${\displaystyle \varphi }$ is the standardnormalpdf
CDF	${\displaystyle \mathrm{\Phi }(y)}$, where ${\displaystyle \mathrm{\Phi }}$ is the standardnormalCDF
Mean	${\displaystyle \xi -\frac{\alpha }{\kappa }\left(e^{{\kappa }^2/2}-1\right)}$
Median	${\displaystyle \xi }$
Variance	${\displaystyle \frac{{\alpha }^2}{{\kappa }^2}{e}^{{\kappa }^2}\left(e^{{\kappa }^2}-1\right)}$
Skewness	${\displaystyle \frac{3e^{{\kappa }^2}-e^{3{\kappa }^2}-2}{(e^{{\kappa }^2}-1{)}^{3/2}}\text{ sign}(\kappa )}$
Excess kurtosis	${\displaystyle e^{4{\kappa }^2}+2e^{3{\kappa }^2}+3e^{2{\kappa }^2}-6}$

Theasymmetric generalized normal distribution is a family of continuous probability distributions in which the shape parameter can be used to introduce asymmetry or skewness.^[15][16] When the shape parameter is zero, the normal distribution results. Positive values of the shape parameter yield left-skewed distributions bounded to the right, and negative values of the shape parameter yield right-skewed distributions bounded to the left. Only when the shape parameter is zero is the density function for this distribution positive over the whole real line: in this case the distribution is anormal distribution, otherwise the distributions are shifted and possibly reversedlog-normal distributions.

Parameters can be estimated viamaximum 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. Theskew normal distribution is another distribution that is useful for modeling deviations from normality due to skew. Other distributions used to model skewed data include thegamma,lognormal, andWeibull distributions, but these do not include the normal distributions as special cases.

Kullback-Leibler divergence (KLD) is a method using for compute the divergence or similarity between two probability density functions.^[17]

Let${\displaystyle P(x)}$ and${\displaystyle Q(x)}$ two generalized Gaussian distributions with parameters${\displaystyle {\alpha }_1,{\beta }_1,{\mu }_1}$ and${\displaystyle {\alpha }_2,{\beta }_2,{\mu }_2}$subject to the constraint${\displaystyle {\mu }_1={\mu }_2=0}$.^[18] Then this divergence is given by:

${\displaystyle \mathrm{K}\mathrm{L}{\mathrm{D}}_{\mathrm{p}\mathrm{d}\mathrm{f}}(P(x)||Q(x))=-\frac{1}{{\beta }_1}+\frac{(\frac{{\alpha }_1}{{\alpha }_2}{)}^{{\beta }_2}\mathrm{\Gamma }(\frac{1+{\beta }_2}{{\beta }_1})}{\mathrm{\Gamma }(\frac{1}{{\beta }_1})}+\log \left(\frac{{\alpha }_2\mathrm{\Gamma }(1+\frac{1}{{\beta }_2})}{{\alpha }_1\mathrm{\Gamma }(1+\frac{1}{{\beta }_1})}\right)}$

The two generalized normal families described here, like theskew normal family, are parametric families that extends the normal distribution by adding a shape parameter. Due to the central role of the normal distribution in probability and statistics, many distributions can be characterized in terms of their relationship to the normal distribution. For example, thelog-normal,folded normal, andinverse normal distributions are defined as transformations of a normally-distributed value, but unlike the generalized normal and skew-normal families, these do not include the normal distributions as special cases.

Actually all distributions with finite variance are in the limit highly related to the normal distribution. The Student-t distribution, theIrwin–Hall distribution and theBates distribution also extend the normal distribution, andinclude in the limit the normal distribution. So there is no strong reason to prefer the "generalized" normal distribution of type 1, e.g. over a combination of Student-t and a normalized extended Irwin–Hall – this would include e.g. the triangular distribution (which cannot be modeled by the generalized Gaussian type 1).

A symmetric distribution which can model both tail (long and short)and center behavior (like flat, triangular or Gaussian) completely independently could be derived e.g. by using X = IH/chi.

TheTukey g- and h-distribution also allows for a deviation from normality, both through skewness and fat tails.^[19]

• Complex normal distribution
• Skew normal distribution

• Continuous distributions
• Normal distribution

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