generalized normal distribution

{{Probability distribution |

name =Symmetric Generalized Normal|

type =density|

pdf_image =File:Generalized normal densities.svg|

cdf_image =File:Generalized normal cdfs.svg|

parameters =$\backslash mu\; \backslash ,$ location (real)
$\backslash alpha\; \backslash ,$ scale (positive, real)
$\backslash beta\; \backslash ,$ shape (positive, real)|

support =$x\; \backslash in\; (-\backslash infty;\; +\backslash infty)\backslash !$|

pdf =$\backslash frac\{\backslash beta\}\{2\backslash alpha\backslash Gamma(1/\backslash beta)\}\; \backslash ;$

e^{-(|x-\mu|/\alpha)^\beta}

$\backslash Gamma$ denotes the gamma function|

cdf =$\backslash frac\{1\}\{2\}\; +\; \backslash text\{sign\}(x\; -\; \backslash mu)\; \backslash frac\{1\}\{2\backslash Gamma(\; 1/\backslash beta\; )\; \}\; \backslash gamma\; \backslash left(1/\backslash beta,\; \backslash left|\; \backslash frac\{x\; -\; \backslash mu\}\{\backslash alpha\}\; \backslash right|^\backslash beta\; \backslash right)$

where $\backslash beta$ is a shape parameter, $\backslash alpha$ is a scale parameter and $\backslash gamma$ is the unnormalized incomplete lower gamma function.|

quantile =$\backslash text\{sign\}(p\; -\; 0.5)\; \backslash left[\; \backslash alpha^\backslash beta\; F^\{-1\}\; \backslash left(2|p\; -\; 0.5|;\; \backslash frac\{1\}\{\backslash beta\}\backslash right)\; \backslash right]^\{1/\backslash beta\}\; +\; \backslash mu$

where $F^\{-1\}\; \backslash left(p;\; a\backslash right)$ is the quantile function of Gamma distribution{{cite web |last1=Griffin |first1=Maryclare |title=Working with the Exponential Power Distribution Using gnorm |url=https://cran.r-project.org/web/packages/gnorm/vignettes/gnormUse.html |website=Github, gnorm package |access-date=26 June 2020}}|

mean =$\backslash mu\; \backslash ,$|

median =$\backslash mu\; \backslash ,$|

mode =$\backslash mu\; \backslash ,$|

variance =$\backslash frac\{\backslash alpha^2\backslash Gamma(3/\backslash beta)\}\{\backslash Gamma(1/\backslash beta)\}$|

skewness =0|

kurtosis =$\backslash frac\{\backslash Gamma(5/\backslash beta)\backslash Gamma(1/\backslash beta)\}\{\backslash Gamma(3/\backslash beta)^2\}-3$|

entropy =$\backslash frac\{1\}\{\backslash beta\}-\backslash log\backslash left[\backslash frac\{\backslash beta\}\{2\backslash alpha\backslash Gamma(1/\backslash beta)\}\backslash right]${{cite journal |last= Nadarajah|first= Saralees|date=September 2005|title= A generalized normal distribution|journal= Journal of Applied Statistics|volume= 32 |issue= 7|pages= 685–694|doi= 10.1080/02664760500079464 |bibcode= 2005JApSt..32..685N|s2cid= 121914682}}|

mgf =|

char =|

}}

The symmetric generalized normal distribution, also known as the exponential power distribution or the generalized error distribution, is a parametric family of symmetric distributions. It includes all normal and Laplace distributions, and as limiting cases it includes all continuous uniform distributions on bounded intervals of the real line.

This family includes the normal distribution when $\backslash textstyle\backslash beta=2$ (with mean $\backslash textstyle\backslash mu$ and variance $\backslash textstyle\; \backslash frac\{\backslash alpha^2\}\{2\}$) and it includes the Laplace distribution when $\backslash textstyle\backslash beta=1$. As $\backslash textstyle\backslash beta\backslash rightarrow\backslash infty$, the density converges pointwise to a uniform density on $\backslash textstyle\; (\backslash mu-\backslash alpha,\backslash mu+\backslash alpha)$.

This family allows for tails that are either heavier than normal (when ) or lighter than normal (when $\backslash beta>2$). It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal ($\backslash textstyle\backslash beta=2$) to the uniform density ($\backslash textstyle\backslash beta=\backslash infty$), and a continuum of symmetric, leptokurtic densities spanning from the Laplace ($\backslash textstyle\backslash beta=1$) to the normal density ($\backslash textstyle\backslash beta=2$).

The shape parameter $\backslash beta$ also controls the peakedness in addition to the tails.

Parameter estimation via maximum likelihood and the method of moments has been studied.{{cite journal |last= Varanasi |first= M.K. |author2=Aazhang, B. |date=October 1989|title= Parametric generalized Gaussian density estimation|journal= Journal of the Acoustical Society of America|volume= 86|issue= 4|pages= 1404–1415|doi= 10.1121/1.398700|bibcode= 1989ASAJ...86.1404V }} The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed.{{cite web |last=Domínguez-Molina |first=J. Armando |author2=González-Farías, Graciela |author2-link=Graciela González Farías |author3=Rodríguez-Dagnino, Ramón M. |title=A practical procedure to estimate the shape parameter in the generalized Gaussian distribution |url=http://www.cimat.mx/reportes/enlinea/I-01-18_eng.pdf |access-date=2009-03-03 |archive-date=2007-09-28 |archive-url=https://web.archive.org/web/20070928061844/http://www.cimat.mx/reportes/enlinea/I-01-18_eng.pdf |url-status=dead }}

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

It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method.{{cite journal |last= Varanasi|first= M.K.|author2=Aazhang B. |year= 1989|title= Parametric generalized Gaussian density estimation|journal= J. Acoust. Soc. Am. |volume= 86|issue= 4|pages= 1404–1415|doi= 10.1121/1.398700|bibcode= 1989ASAJ...86.1404V}}{{cite journal |last= Do |first= M.N.|author2=Vetterli, M. |date=February 2002|title= Wavelet-based Texture Retrieval Using Generalised Gaussian Density and Kullback-Leibler Distance|journal= IEEE Transactions on Image Processing|volume= 11|issue= 2|pages= 146–158|url= http://infoscience.epfl.ch/record/33839|doi= 10.1109/83.982822|pmid= 18244620|bibcode= 2002ITIP...11..146D}} With $\backslash mu$ initially set to the sample first moment $m\_1$,

$\backslash textstyle\backslash beta$ is estimated by using a Newton–Raphson iterative procedure, starting from an initial guess of $\backslash textstyle\backslash beta=\backslash textstyle\backslash beta\_0$,

:$\backslash beta\; \_0\; =\; \backslash frac\{m\_1\}\{\backslash sqrt\{m\_2\}\},$

where

:$m\_1=\{1\; \backslash over\; N\}\; \backslash sum\_\{i=1\}^N\; |x\_i|,$

is the first statistical moment of the absolute values and $m\_2$ is the second statistical moment. The iteration is

:$\backslash beta\_\{i+1\}\; =\; \backslash beta\_\{i\}\; -\; \backslash frac\{g(\backslash beta\; \_\{i\})\}\{g\text{'}(\backslash beta\_\{i\})\}\; ,$

where

:$g(\backslash beta)=\; 1\; +\; \backslash frac\{\backslash psi(1/\backslash beta)\}\{\backslash beta\}\; -\; \backslash frac\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\backslash beta\; \backslash log|x\_i-\backslash mu|\; \}\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\backslash beta\}\; +\; \backslash frac\{\backslash log(\; \backslash frac\{\backslash beta\}\{N\}\; \backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\backslash beta)\}\{\backslash beta\}\; ,$

and

: $$

\begin{align}

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{align}

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

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

:$\backslash min\_\backslash mu\; =\; \backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\backslash beta$

Finally $\backslash textstyle\backslash alpha$ is evaluated as

:$\backslash alpha\; =\; \backslash left(\; \backslash frac\{\backslash beta\}\{N\}\; \backslash sum\_\{i=1\}^N|x\_i-\backslash mu|^\backslash beta\backslash right)^\{1/\backslash beta\}\; .$

For $\backslash beta\; \backslash leq\; 1$, median is a more appropriate estimator of $\backslash mu$ . Once $\backslash mu$ is estimated, $\backslash beta$ and $\backslash alpha$ can be estimated as described above.{{Cite journal|last1=Varanasi|first1=Mahesh K.|last2=Aazhang|first2=Behnaam|date=1989-10-01|title=Parametric generalized Gaussian density estimation|journal=The Journal of the Acoustical Society of America|volume=86|issue=4|pages=1404–1415|doi=10.1121/1.398700|bibcode=1989ASAJ...86.1404V |issn=0001-4966}}

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.

{{cite journal

|last = Liang

|first = Faming

|author2 = Liu, Chuanhai

|author3 = Wang, Naisyin | author3-link = Naisyin Wang

|date = April 2007

|title = A robust sequential Bayesian method for identification of differentially expressed genes

|journal = Statistica Sinica

|volume = 17

|issue = 2

|pages = 571–597

|url = http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=17&num=2&art=8

|access-date = 2009-03-03

|archive-url = https://web.archive.org/web/20071009233343/http://www3.stat.sinica.edu.tw/statistica/password.asp?vol=17&num=2&art=8

|archive-date = 2007-10-09

}}

{{cite book |title= Bayesian Inference in Statistical Analysis |last= Box |first= George E. P.|author-link= George E. P. Box |author2-link=George Tiao|author2=Tiao, George C. |year= 1992 |publisher= Wiley|location= New York|isbn= 978-0-471-57428-6}} 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. It finds uses in plasma physics under the name of Langdon Distribution resulting from inverse bremsstrahlung.{{cite thesis |degree=PhD |title=Electron velocity distribution functions and Thomson scattering |last=Milder |first=Avram L. |year=2021 |publisher=University of Rochester |hdl=1802/36536 |hdl-access=free}}

In a linear regression problem modeled as $y\; \backslash sim\; \backslash mathrm\{GeneralizedNormal\}(X\backslash cdot\backslash theta,\; \backslash alpha,\; p)$, the MLE will be the $\backslash arg\backslash min\_\{\backslash theta\}\backslash |X\backslash cdot\backslash theta-y\backslash |\_p$ where the p-norm is used.

Let $X\_\backslash beta$ be zero mean generalized Gaussian distribution of shape $\backslash beta$ and scaling parameter $\backslash alpha$ . The moments of $X\_\backslash beta$ exist and are finite for any k greater than −1. For any non-negative integer k, the plain central moments are

: $$

\operatorname{E}\left[X^k_\beta\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, $\backslash beta$ 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:

: $$

\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 $\backslash mathfrak\{N\}\_\backslash beta(\backslash nu)$ is the Stable count distribution and $V\_\{\backslash beta\}(s)$

is the Stable vol distribution.

The probability density function of the symmetric generalized normal distribution is a positive-definite function for $\backslash beta\; \backslash in\; (0,2]$.{{cite journal

|last = Dytso

|first = Alex

|author2 = Bustin, Ronit

|author3 = Poor, H. Vincent

|author4 = Shamai, Shlomo

|date = 2018

|title = Analytical properties of generalized Gaussian distributions

|journal = Journal of Statistical Distributions and Applications

|volume = 5

|issue = 1

|page = 6

|doi=10.1186/s40488-018-0088-5

|doi-access= free

}}

{{cite journal

|last = Bochner

|first = Salomon

|date = 1937

|title = Stable laws of probability and completely monotone functions

|journal = Duke Mathematical Journal

|volume = 3

|issue = 4

|pages = 726–728

|url = https://jsdajournal.springeropen.com/articles/10.1186/s40488-018-0088-5

|doi=10.1215/s0012-7094-37-00360-0

}}

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

The multivariate generalized normal distribution, i.e. the product of $n$ exponential power distributions with the same $\backslash beta$ and $\backslash alpha$ parameters, is the only probability density that can be written in the form $p(\backslash mathbf\; x)=g(\backslash |\backslash mathbf\; x\backslash |\_\backslash beta)$ and has independent marginals.{{cite journal |last= Sinz|first= Fabian|author2=Gerwinn, Sebastian |author3=Bethge, Matthias

|date=May 2009|title=Characterization of the p-Generalized Normal Distribution. |journal=Journal of Multivariate Analysis|volume= 100|issue= 5|pages= 817–820|doi=10.1016/j.jmva.2008.07.006|doi-access=free}} The results for the special case of the Multivariate normal distribution is originally attributed to Maxwell.{{cite journal |last= Kac|first= M.|year= 1939|title=On a characterization of the normal distribution|journal=American Journal of Mathematics|volume= 61|issue= 3|pages= 726–728|doi= 10.2307/2371328 |jstor= 2371328}}

{{Probability distribution |

name =Asymmetric Generalized Normal|

type =density|

pdf_image =File:Generalized normal densities 2.svg|

cdf_image =File:Generalized normal cdfs 2.svg|

parameters =$\backslash xi\; \backslash ,$ location (real)
$\backslash alpha\; \backslash ,$ scale (positive, real)
$\backslash kappa\; \backslash ,$ shape (real)|

support =$x\; \backslash in\; (-\backslash infty,\backslash xi+\backslash alpha/\backslash kappa)\; \backslash text\{\; if\; \}\; \backslash kappa>0$
$x\; \backslash in\; (-\backslash infty,\backslash infty)\; \backslash text\{\; if\; \}\; \backslash kappa=0$
|

pdf =$\backslash frac\{\backslash phi(y)\}\{\backslash alpha-\backslash kappa(x-\backslash xi)\}$, where

$\backslash phi$ is the standard normal pdf|

cdf =$\backslash Phi(y)$, where

$\backslash Phi$ is the standard normal CDF|

mean =$\backslash xi\; -\; \backslash frac\{\backslash alpha\}\{\backslash kappa\}\; \backslash left(\; e^\{\backslash kappa^2/2\}\; -\; 1\; \backslash right)$|

median =$\backslash xi\; \backslash ,$|

mode =|

variance =$\backslash frac\{\backslash alpha^2\}\{\backslash kappa^2\}\; e^\{\backslash kappa^2\}\; \backslash left(\; e^\{\backslash kappa^2\}\; -\; 1\; \backslash right)$|

skewness =$\backslash frac\{3\; e^\{\backslash kappa^2\}\; -\; e^\{3\; \backslash kappa^2\}\; -\; 2\}\{(e^\{\backslash kappa^2\}\; -\; 1)^\{3/2\}\}\; \backslash text\{\; sign\}(\backslash kappa)$|

kurtosis =$e^\{4\; \backslash kappa^2\}\; +\; 2\; e^\{3\; \backslash kappa^2\}\; +\; 3\; e^\{2\; \backslash kappa^2\}\; -\; 6$|

entropy =|

mgf =|

char =|

}}

{{distinguish|Skew normal distribution}}

The asymmetric generalized normal distribution is a family of continuous probability distributions in which the shape parameter can be used to introduce asymmetry or skewness.Hosking, J.R.M., Wallis, J.R. (1997) Regional frequency analysis: an approach based on L-moments, Cambridge University Press. {{ISBN|0-521-43045-3}}. Section A.8[http://www.cran.r-project.org/web/packages/lmomco/lmomco.pdf Documentation for the lmomco R package] 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 a normal distribution, otherwise the distributions are shifted and possibly reversed log-normal distributions.

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.

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

{{cite journal

|last = Kullback

|first = S.

|author2 = Leibler, R.A.

|date = 1951

|title = On information and sufficiency

|journal = The Annals of Mathematical Statistics

|volume = 22

|issue = 1

|pages = 79–86

|doi=10.1214/aoms/1177729694

|doi-access= free

}}

Let $P(x)$ and $Q(x)$ two generalized Gaussian distributions with parameters $\backslash alpha\_1,\; \backslash beta\_1,\; \backslash mu\_1$ and $\backslash alpha\_2,\; \backslash beta\_2,\; \backslash mu\_2$

subject to the constraint $\backslash mu\_1\; =\; \backslash mu\_2\; =\; 0$.

{{cite journal

|last = Quintero-Rincón

|first = A.

|author2 = Pereyra, M.

|author3 = D’Giano, C.

|author4 = Batatia, H.

|author5 = Risk, M.

|date = 2017

|title = A visual EEG epilepsy detection method based on a wavelet statistical representation and the Kullback-Leibler divergence

|journal = IFMBE Proceedings

|volume = 60

|pages = 13–16

|doi=10.1007/978-981-10-4086-3_4

|doi-access= free

|hdl= 11336/77054

|hdl-access= free

}} Then this divergence is given by:

:$KLD\_\{pdf\}(P(x)||Q(x))\; =\; -\backslash frac\{1\}\{\backslash beta\_1\}\; +\; \backslash frac\{(\backslash frac\{\backslash alpha\_1\}\{\backslash alpha\_2\})^\{\backslash beta\_2\}\backslash Gamma\; (\backslash frac\{1+\backslash beta\_2\}\{\backslash beta\_1\})\}\{\backslash Gamma\; (\backslash frac\{1\}\{\backslash beta\_1\})\}\; +\; \backslash log\; \backslash left(\backslash frac\{\backslash alpha\_2\; \backslash Gamma(1+\; \backslash frac\{1\}\{\backslash beta\_2\})\}\{\backslash alpha\_1\; \backslash Gamma(1+\; \backslash frac\{1\}\{\backslash beta\_1\})\}\backslash right)$

The two generalized normal families described here, like the skew 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, the log-normal, folded normal, and inverse 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, the Irwin–Hall distribution and the Bates distribution also extend the normal distribution, and include 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.

The Tukey g- and h-distribution also allows for a deviation from normality, both through skewness and fat tails.[https://academic.oup.com/jrssig/article/16/3/12/7037766?login=false The Tukey g-and-h Distribution],

Yuan Yan, Marc G. Genton

Significance, Volume 16, Issue 3, June 2019, Pages 12–13, {{doi|10.1111/j.1740-9713.2019.01273.x}}

• Complex normal distribution
• Skew normal distribution

{{Reflist}}

{{ProbDistributions|continuous}}

{{Statistics|hide}}

{{DEFAULTSORT:Generalized Normal Distribution}}

Category:Continuous distributions