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Generalized Normal Distribution

docs/tutorial:stats:continuous_gennorm

This distribution is also known as the exponential power distribution. It has a single shape parameter $β > 0$ . It reduces to a number of common distributions.

Functions

\begin{eqnarray*} f\left(x; \beta\right) & = &\frac{\beta}{2\Gamma(1/\beta)} e^{-\left|x\right|^{\beta}} \end{eqnarray*}

\begin{eqnarray*} F\left(x; \beta\right) & = & \frac{1}{2} + \mathrm{sgn}\left(x\right) \frac{\gamma\left(1/\beta, x^{\beta}\right)}{2\Gamma\left(1/\beta\right)} \end{eqnarray*}

$γ$ is the lower incomplete gamma function. $γ (s, x) = \int_{0}^{x} t^{s - 1} e^{- t} d t$ .

\begin{eqnarray*} h\left[X; \beta\right] = \frac{1}{\beta} - \log\left(\frac{\beta}{2\Gamma\left(1/\beta\right)}\right)\end{eqnarray*}

Moments

Special Cases

Laplace distribution ( $β = 1$ )
Normal distribution with $μ_{2} = 1/2$ ( $β = 2$ )
Uniform distribution over the interval $[- 1, 1]$ ( $β \to \infty$ )

Sources

https://en.wikipedia.org/wiki/Generalized_normal_distribution#Version_1
https://en.wikipedia.org/wiki/Incomplete_gamma_function#Lower_incomplete_Gamma_function

Implementation: scipy.stats.gennorm