Kaniadakis Gaussian distribution

Short description: Continuous probability distribution

κ-Gaussian distribution

Probability density function
Cumulative distribution function
Parameters	$0<\kappa <1$ shape (real) $\beta >0$ rate (real)
Support	$x\in \mathbb{ℝ}$
PDF	$Z_{\kappa }{\exp }_{\kappa }(-\beta x^2);Z_{\kappa }=\sqrt{\frac{2\beta \kappa }{\pi }}(1+\frac{1}{2}\kappa )\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}$
CDF	$\frac{1}{2}+\frac{1}{2}{\text{<mrow data-mjx-texclass="ORD">erf</mrow>}}_{\kappa }\left(\sqrt{\beta }x\right)$
Mean	$0$
Median	$0$
Mode	$0$
Variance	${\sigma }_{\kappa }^2=\frac{1}{\beta }\frac{2+\kappa }{2-\kappa }\frac{4\kappa }{4-9{\kappa }^2}{\left[\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}\right]}^2$
Skewness	$0$
Kurtosis	$3[\frac{\sqrt{\pi }{Z}_{\kappa }}{2{\beta }^{2/3}{\sigma }_{\kappa }^4}\frac{(2\kappa {)}^{-5/2}}{1+\frac{5}{2}\kappa }\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{5}{4})}-1]$

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:

$f_{{}_{\kappa }}(x)=Z_{\kappa }{\exp }_{\kappa }(-\beta x^2)$

where $|\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $\beta >0$ is the scale parameter, and

$Z_{\kappa }=\sqrt{\frac{2\beta \kappa }{\pi }}(1+\frac{1}{2}\kappa )\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}$

is the normalization constant.

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

Cumulative distribution function

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

$F_{\kappa }(x)=\frac{1}{2}+\frac{1}{2}{\text{<mrow data-mjx-texclass="ORD">erf</mrow>}}_{\kappa }\left(\sqrt{\beta }x\right)$

where

${\text{<mrow data-mjx-texclass="ORD">erf</mrow>}}_{\kappa }(x)=(2+\kappa )\sqrt{\frac{2\kappa }{\pi }}\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}\underset{0}{\overset{x}{\int }}{\exp }_{\kappa }(-t^2)dt$

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

$\text{<mrow data-mjx-texclass="ORD">erf</mrow>}(x)$

as

$\kappa \to 0$

.

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:

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

Kurtosis

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

$\mathrm{Kurt}[X]=\mathrm{E}\left[\frac{X^4}{{\sigma }_{\kappa }^4}\right]$

which can be written as

$\mathrm{Kurt}[X]=\frac{2Z_{\kappa }}{{\sigma }_{\kappa }^4}\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^4{\exp }_{\kappa }(-\beta x^2)dx$

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

$\mathrm{Kurt}[X]=\frac{3\sqrt{\pi }{Z}_{\kappa }}{2{\beta }^{2/3}{\sigma }_{\kappa }^4}\frac{|2\kappa {|}^{-5/2}}{1+\frac{5}{2}|\kappa |}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{5}{4})}$

or

$\mathrm{Kurt}[X]=\frac{3{\beta }^{11/6}\sqrt{2\kappa }}{2}\frac{|2\kappa {|}^{-5/2}}{1+\frac{5}{2}|\kappa |}(1+\frac{1}{2}\kappa ){\left(\frac{2-\kappa }{2+\kappa }\right)}^2{\left(\frac{4-9{\kappa }^2}{4\kappa }\right)}^2{\left[\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}\right]}^3\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{5}{4})}$

κ-Error function

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

${\mathrm{erf}}_{\kappa }(x)=(2+\kappa )\sqrt{\frac{2\kappa }{\pi }}\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{1}{4})}\underset{0}{\overset{x}{\int }}{\exp }_{\kappa }(-t^2)dt$

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.

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

External links

• Kaniadakis Statistics on arXiv.org