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.

Definitions

Probability density function

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

: $f_(x) = Z_\kappa \exp_\kappa(-\beta x^2)$
where $, \kappa, < 1$ is the entropic index associated with the Kaniadakis entropy, $\beta > 0$ is the scale parameter, and
:$Z_\kappa = \sqrt \Bigg( 1 + \frac\kappa \Bigg) \frac$
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

$F_\kappa(x) = \frac + \frac \textrm_\kappa \big( \sqrt x\big)$

where

$\textrm_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt \frac \int_0^x \exp_\kappa(-t^2 ) dt$

is the Kaniadakis ''κ''-Error function, which is a generalization of the ordinary Error function $\textrm(x)$ as $\kappa \rightarrow 0$.

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 $\kappa < 2/3$ and is given by:

: \operatorname = \sigma_\kappa^2 = \frac \frac \frac \left frac\right2

Kurtosis

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

: \operatorname = \operatorname\left frac\right
which can be written as

\operatorname = \frac \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:

\operatorname = \frac \frac \frac

or

\operatorname = \frac \frac \Bigg( 1 + \frac\kappa \Bigg) \left(\frac \right)^2 \left( \frac \right)^2 \left frac\right3 \frac

κ-Error function

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

:$\operatorname_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt \frac \int_0^x \exp_\kappa(-t^2 ) dt$

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

For a random variable 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/math>.

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.

See also

* Giorgio Kaniadakis
* Kaniadakis statistics
* Kaniadakis distribution
* Kaniadakis κ-Exponential distribution
* Kaniadakis κ-Gamma distribution
* Kaniadakis κ-Weibull distribution
* Kaniadakis κ-Logistic distribution
* Kaniadakis κ-Erlang distribution

References

{{Reflist

External links

Kaniadakis Statistics on arXiv.org

Probability distributions
Mathematical and quantitative methods (economics)