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,[1] geophysics,[2] astrophysics, among many others.

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

[3] The general form of the centered Kaniadakis κ-Gaussian probability density function is:[3] where

κ

is the entropic index associated with the Kaniadakis entropy,

is the scale parameter, and is the normalization constant.

The standard Normal distribution is recovered in the limit

κ → 0.

The cumulative distribution function of κ-Gaussian distribution is given by

κ

κ

κ

2 + κ

2 κ

2 κ

2 κ

κ

{\displaystyle {\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

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

The variance is finite for

and is given by: The kurtosis of the centered κ-Gaussian distribution may be computed thought:

{\displaystyle \operatorname {Kurt} [X]={\frac {2Z_{\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:

The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:[3] 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

, κ-Error function means the probability that X falls in the interval

The κ-Gaussian distribution has been applied in several areas, such as: