Kaniadakis Gaussian distribution

The Kaniadakis 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.

Definitions

Probability density function

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

Cumulative distribution function

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

where

is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function as .

Properties

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 and is given by:

Kurtosis

The kurtosis of the centered κ-Gaussian distribution may be computed thought:
which can be written as

Thus, the kurtosis of the centered κ-Gaussian distribution is given by:

κ-Error function

The Kaniadakis κ-Error function is a one-parameter generalization of the ordinary error function defined as:
Although the error function cannot be expressed in terms of elementary functions, numerical approximations are commonly employed.
For a random variable 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 .

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.