Kaniadakis Gaussian distribution

The Kaniadakis Gaussian distribution (also known as κ-Gaussian distribution) is a probability distribution which arises as a generalization of the Gaussian distribution from the maximization of the Kaniadakis entropy under appropriated constraints. It is one example of a Kaniadakis κ-distribution. The κ-Gaussian distribution has been applied successfully for describing several complex systems in economy, geophysics, astrophysics, among many others.

The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution.

Probability density function
The general form of the centered Kaniadakis κ-Gaussian probability density function is:



f_{_{\kappa}}(x) = Z_\kappa \exp_\kappa(-\beta x^2) $$ where $$|\kappa| < 1$$ is the entropic index associated with the Kaniadakis entropy, $$\beta > 0$$ is the scale parameter, and

Z_\kappa = \sqrt{\frac{2 \beta \kappa}{ \pi } } \Bigg( 1 + \frac{1}{2}\kappa \Bigg) \frac{ \Gamma \Big( \frac{1}{2 \kappa} + \frac{1}{4}\Big)}{ \Gamma \Big( \frac{1}{2 \kappa} - \frac{1}{4}\Big) } $$ is the normalization constant.

The standard Normal distribution is recovered in the limit $$\kappa \rightarrow 0.$$

Cumulative distribution function
The cumulative distribution function of κ-Gaussian distribution is given by $$F_\kappa(x) = \frac{1}{2} + \frac{1}{2} \textrm{erf}_\kappa \big( \sqrt{\beta} x\big)$$ where $$\textrm{erf}_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma\Big( \frac{1}{2\kappa} + \frac{1}{4}  \Big)}{ \Gamma\Big( \frac{1}{2\kappa} - \frac{1}{4}  \Big) }  \int_0^x \exp_\kappa(-t^2 ) dt$$ is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function $$\textrm{erf}(x)$$ as $$\kappa \rightarrow 0$$.

Moments, mean and variance
The centered κ-Gaussian distribution has a moment of odd order equal to zero, including the mean.

The variance is finite for $$\kappa < 2/3$$ and is given by:


 * $$\operatorname{Var}[X] = \sigma_\kappa^2 = \frac{1}{\beta} \frac{2 + \kappa}{2 - \kappa} \frac{4\kappa}{4 - 9 \kappa^2 } \left[\frac{\Gamma \left( \frac{1}{2\kappa} + \frac{1}{ 4 }\right)}{\Gamma \left( \frac{1}{2\kappa} - \frac{1}{ 4 }\right)}\right]^2 $$

Kurtosis
The kurtosis of the centered κ-Gaussian distribution may be computed thought:


 * $$\operatorname{Kurt}[X] = \operatorname{E}\left[\frac{X^4}{\sigma_\kappa^4}\right] $$

which can be written as"$\operatorname{Kurt}[X] = \frac{2 Z_\kappa}{\sigma_\kappa^4} \int_0^\infty x^4 \, \exp_\kappa \left( -\beta x^2 \right) dx $"Thus, the kurtosis of the centered κ-Gaussian distribution is given by:"2 \kappa"or"2 \kappa"

κ-Error function
The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:


 * $$\operatorname{erf}_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma\Big( \frac{1}{2\kappa} + \frac{1}{4}  \Big)}{ \Gamma\Big( \frac{1}{2\kappa} - \frac{1}{4}  \Big) }  \int_0^x \exp_\kappa(-t^2 )

dt$$

Although the error function cannot be expressed in terms of elementary functions, numerical approximations are commonly employed.

For a random variable $X$ distributed according to a κ-Gaussian distribution with mean 0 and standard deviation $$\sqrt \beta$$, κ-Error function means the probability that X falls in the interval $$[-x, \, x]$$.

Applications
The κ-Gaussian distribution has been applied in several areas, such as:
 * In economy, the κ-Gaussian distribution has been applied in the analysis of financial models, accurately representing the dynamics of the processes of extreme changes in stock prices.
 * In inverse problems, Error laws in extreme statistics are robustly represented by κ-Gaussian distributions.
 * In astrophysics, stellar-residual-radial-velocity data have a Gaussian-type statistical distribution, in which the K index presents a strong relationship with the stellar-cluster ages.
 * In nuclear physics, the study of Doppler broadening function in nuclear reactors is well described by a κ-Gaussian distribution for analyzing the neutron-nuclei interaction.
 * In cosmology, for interpreting the dynamical evolution of the Friedmann–Robertson–Walker Universe.
 * In plasmas physics, for analyzing the electron distribution in electron-acoustic double-layers and the dispersion of Langmuir waves.