Kaniadakis Gaussian distribution

κ-Gaussian distribution

Probability density function
Cumulative distribution function
Parameters	[math]\displaystyle{ 0 \lt \kappa \lt 1 }[/math] shape (real) [math]\displaystyle{ \beta\gt 0 }[/math] rate (real)
Support	[math]\displaystyle{ x \in \mathbb{R} }[/math]
PDF	[math]\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) } }[/math]
CDF	[math]\displaystyle{ \frac{1}{2} + \frac{1}{2} \textrm{erf}_\kappa \big( \sqrt{\beta} x\big)\ }[/math]
Mean	[math]\displaystyle{ 0 }[/math]
Median	[math]\displaystyle{ 0 }[/math]
Mode	[math]\displaystyle{ 0 }[/math]
Variance	[math]\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 }[/math]
Skewness	[math]\displaystyle{ 0 }[/math]
Kurtosis	[math]\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] }[/math]

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:

[math]\displaystyle{ f_{_{\kappa}}(x) = Z_\kappa \exp_\kappa(-\beta x^2) }[/math]

where [math]\displaystyle{ |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy, [math]\displaystyle{ \beta \gt 0 }[/math] is the scale parameter, and

[math]\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) } }[/math]

is the normalization constant.

The standard Normal distribution is recovered in the limit [math]\displaystyle{ \kappa \rightarrow 0. }[/math]

Cumulative distribution function

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

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

where

[math]\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 }[/math]

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

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 [math]\displaystyle{ \kappa \lt 2/3 }[/math] and is given by:

[math]\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 }[/math]

Kurtosis

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

[math]\displaystyle{ \operatorname{Kurt}[X] = \operatorname{E}\left[\frac{X^4}{\sigma_\kappa^4}\right] }[/math]

which can be written as

[math]\displaystyle{ \operatorname{Kurt}[X] = \frac{2 Z_\kappa}{\sigma_\kappa^4} \int_0^\infty x^4 \, \exp_\kappa \left( -\beta x^2 \right) dx }[/math]

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

[math]\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)} }[/math]

or

[math]\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)} }[/math]

κ-Error function

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

[math]\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 }[/math]

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

External links

• Kaniadakis Statistics on arXiv.org