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DISTRIBUTIONS / CONTINUOUS / GENERALIZED NORMAL

Generalized Normal distribution

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The Generalized Normal, also called the exponential-power or Subbotin distribution, extends the normal by changing the power of distance from the centre. It is symmetric but controls tail weight explicitly.

Feature Reading
Support It spans the real line. mu is the centre, alpha the scale, and beta the power: beta=2 gives normal and beta=1 gives Laplace.
Shape It connects normal and Laplace along a continuum of tails. Powers below 2 generally put more mass in the tails than normal; powers above 2 concentrate more tightly around the centre.

Origin and terminology

The family was studied in several forms by Subbotin and became popular as the generalized error distribution in robust statistics and time series.

In real models

  • symmetric errors and robust regression
  • time-series residuals, images, and signals with non-Gaussian tails

Comparisons

Do not read beta as a probability; it is a shape power, and alpha is not automatically the standard deviation.

A continuous dial between Laplace, Normal, and a box

The exponential-power Generalized Normal changes the exponent applied to distance from centre. At beta=1 it recovers Laplace. At beta=2 it gives a Normal law after translating the scale. As beta tends to infinity, the shape approaches a symmetric Uniform distribution on an interval.

It is also called the Subbotin distribution. Small values create a sharp peak and heavy tails; large values flatten the centre and lighten the tails. The family remains symmetric, so it cannot explain skewness even when it fits kurtosis well. In images, signals, and robust error models, separating asymmetry from tail weight prevents asking beta to solve two different problems.

Decision guide

A good candidate when: errors are symmetric but their peak and kurtosis do not agree with a Normal model.

Compare it with: Student’s t for polynomial tails and an asymmetric family for skewed residuals. Generalized Normal changes kurtosis, not asymmetry.

References

  • SciPy reference: scipy.stats.gennorm — definition and parameterization
  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1994). Continuous Univariate Distributions, 2nd ed., Vol. 1. Wiley.
  • Subbotin, M. T. (1923). On the law of frequency of errors. Matematicheskii Sbornik, 31, 296–301.
  • Box, G. E. P. & Tiao, G. C. (1973). Bayesian Inference in Statistical Analysis. Wiley.

Generalized Normal Distribution: equations and calculator

Distribution defintion

X∼GeneralizedNormal(β,μ,α)X\sim\mathrm{GeneralizedNormal}\left(\beta,\mu,\alpha\right)

Distribution domain

x∈(−∞,+∞)x\in\left(-\infty,+\infty\right)

Parameters domain and parameters constraints

β∈R+,μ∈R,α∈R+\beta\in\mathbb{R}^{+},\mu\in\mathbb{R},\alpha\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=12+sign(x−μ)2Γ(1/β)γ(1/β,∣x−μα∣β)=12+sign(x−μ)2P(1/β,∣x−μα∣β)F_{X}\left(x\right)=\frac{1}{2}+\frac{\text{sign}(x-\mu)}{2\Gamma( 1/\beta ) } \gamma\left(1/\beta,\left|\frac{x-\mu}{\alpha}\right|^\beta\right)=\frac{1}{2}+\frac{\text{sign}(x-\mu)}{2} \text{P}\left(1/\beta,\left|\frac{x-\mu}{\alpha}\right|^\beta\right)

Probability density function

fX(x)=β2αΓ(1/β)exp⁡(−(∣x−μ∣α)β)f_{X}\left(x\right)=\frac{\beta}{2\alpha\Gamma(1/\beta)}\exp\left(-\left(\frac{|x-\mu|}{\alpha}\right)^\beta\right)

Percent point function/Sample

FX−1(u)=sign(u−12)[αβP−1(1β,2∣u−12∣)]1/β+μF^{-1}_{X}\left(u\right)=\text{sign}(u-\frac{1}{2}) \left[\alpha^\beta \text{P}^{-1}\left(\frac{1}{\beta},2|u-\frac{1}{2}|\right)\right]^{1/\beta}+\mu

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx={0if k is oddαkΓ(k+1β)/Γ(1β)if k is even\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\left\{\begin{array}{cl}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{array} \right.

Parametric mean

Mean(X)=μ+αμ1′=μ\mathrm{Mean}(X)=\mu+\alpha\mu'_{1}=\mu

Parametric variance

Variance(X)=α2(μ2′−μ1′2)=α2Γ(3/β)Γ(1/β)\mathrm{Variance}(X)=\alpha^2(\mu'_{2}-\mu'^{2}_{1})=\frac{\alpha^2\Gamma(3/\beta)}{\Gamma(1/\beta)}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=Γ(5/β)Γ(1/β)Γ(3/β)2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{\Gamma(5/\beta)\Gamma(1/\beta)}{\Gamma(3/\beta)^2}

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ\mathrm{Mode}(X)=\mu

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
α:Scale parameter\alpha:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}