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

κ-Gaussian distribution

Probability density function
Cumulative distribution function
Parameters	${\displaystyle 0<\kappa <1}$ shape (real) ${\displaystyle \beta >0}$ rate (real)
Support	${\displaystyle x\in \mathbb {R} }$
PDF	${\displaystyle Z_{\kappa }\exp _{\kappa }(-\beta x^{2})\,\,\,;\,\,\,Z_{\kappa }={\sqrt {\frac {2\beta \kappa }{\pi }}}{\Bigg (}1+{\frac {1}{2}}\kappa {\Bigg )}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}}$
CDF	${\displaystyle {\frac {1}{2}}+{\frac {1}{2}}{\textrm {erf}}_{\kappa }{\big (}{\sqrt {\beta }}x{\big )}\ }$
Mean	${\displaystyle 0}$
Median	${\displaystyle 0}$
Mode	${\displaystyle 0}$
Variance	${\displaystyle \sigma _{\kappa }^{2}={\frac {1}{\beta }}{\frac {2+\kappa }{2-\kappa }}{\frac {4\kappa }{4-9\kappa ^{2}}}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\right]^{2}}$
Skewness	${\displaystyle 0}$
Excess kurtosis	${\displaystyle 3\left[{\frac {{\sqrt {\pi }}Z_{\kappa }}{2\beta ^{2/3}\sigma _{\kappa }^{4}}}{\frac {(2\kappa )^{-5/2}}{1+{\frac {5}{2}}\kappa }}{\frac {\Gamma \left({\frac {1}{2\kappa }}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{2\kappa }}+{\frac {5}{4}}\right)}}-1\right]}$

The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution.^[3]

Definitions

Probability density function

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

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

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

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

is the normalization constant.

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

Cumulative distribution function

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

${\displaystyle F_{\kappa }(x)={\frac {1}{2}}+{\frac {1}{2}}{\textrm {erf}}_{\kappa }{\big (}{\sqrt {\beta }}x{\big )}}$

where

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

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

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

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

Kurtosis

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

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

which can be written as

${\displaystyle \operatorname {Kurt} [X]={\frac {2Z_{\kappa }}{\sigma _{\kappa }^{4}}}\int _{0}^{\infty }x^{4}\,\exp _{\kappa }\left(-\beta x^{2}\right)dx}$

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

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

or

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

κ-Error function

κ-Error function
Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.
General information
General definition	${\displaystyle \operatorname {erf} _{\kappa }(x)={\Big (}2+\kappa {\Big )}{\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\int _{0}^{x}\exp _{\kappa }(-t^{2})dt}$
Fields of application	Probability, thermodynamics
Domain, codomain and image
Domain	${\displaystyle \mathbb {C} }$
Image	${\displaystyle \left(-1,1\right)}$
Specific features
Root	${\displaystyle 0}$
Derivative	${\displaystyle {\frac {d}{dx}}\operatorname {erf} _{\kappa }(x)=\left(2+\kappa \right){\sqrt {\frac {2\kappa }{\pi }}}{\frac {\Gamma \left({\frac {1}{2\kappa }}+{\frac {1}{4}}\right)}{\Gamma \left({\frac {1}{2\kappa }}-{\frac {1}{4}}\right)}}\exp _{\kappa }(-x^{2})}$

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

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

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 ${\displaystyle {\sqrt {\beta }}}$, κ-Error function means the probability that X falls in the interval ${\displaystyle [-x,\,x]}$.

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.^[4]
• In inverse problems, Error laws in extreme statistics are robustly represented by κ-Gaussian distributions.^[2][5][6]
• 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.^[7][8]
• 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.^[9][10]
• 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^[11] and the dispersion of Langmuir waves.^[12]

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