Kaniadakis Gaussian distribution

{{Infobox probability distribution|name=κ-Gaussian distribution|type=density|parameters= $0 < \kappa < 1$ shape (real)
$\beta> 0$ rate (real)|support= $x \in \mathbb{R}$ |pdf= $Z_\kappa \exp_\kappa(-\beta x^2) \, \, \, ; \, \, \, Z_\kappa = \sqrt{\frac{2 \beta \kappa}{ \pi } } \Bigg( 1 + \frac{1}{2}\kappa \Bigg)
\frac{ \Gamma \Big( \frac{1}{2 \kappa} + \frac{1}{4}\Big)}{ \Gamma \Big( \frac{1}{2 \kappa} - \frac{1}{4}\Big) }$ |cdf= $\frac{1}{2} + \frac{1}{2} \textrm{erf}_\kappa \big( \sqrt{\beta} x\big)\$ |mode= $0$ |median= $0$ |variance= $\sigma_\kappa^2 = \frac{1}{\beta} \frac{2 + \kappa}{2 - \kappa} \frac{4\kappa}{4 - 9 \kappa^2 } \left[ \frac{\Gamma \Big( \frac{1}{2\kappa} + \frac{1}{ 4 }\Big)}{\Gamma \Big( \frac{1}{2\kappa} - \frac{1}{ 4 }\Big)} \right]^2$ |pdf_image=File:Kaniadakis Gaussian Distribution Type II pdf.png|cdf_image=File:Kaniadakis Gaussian Distribution Type II cdf.png|mean= $0$ |skewness= $0$ |kurtosis= $3\left[\frac{\sqrt{\pi} Z_\kappa}{ 2 \beta^{2/3} \sigma_\kappa^4 } \frac{ (2 \kappa)^{-5/2} }{1+\frac{5}{2} \kappa } \frac{\Gamma \left( \frac{1}{ 2 \kappa } - \frac{5}{4} \right)}{\Gamma \left( \frac{1}{ 2 \kappa } + \frac{5}{4} \right)} -1 \right]$ }}

The 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,{{Cite journal |last1=Moretto |first1=Enrico |last2=Pasquali |first2=Sara |last3=Trivellato |first3=Barbara |date=2017 |title=A non-Gaussian option pricing model based on Kaniadakis exponential deformation |url=http://link.springer.com/10.1140/epjb/e2017-80112-x |journal=The European Physical Journal B |language=en |volume=90 |issue=10 |pages=179 |doi=10.1140/epjb/e2017-80112-x |bibcode=2017EPJB...90..179M |s2cid=254116243 |issn=1434-6028}} geophysics,{{Cite journal |last1=da Silva |first1=Sérgio Luiz E. F. |last2=Carvalho |first2=Pedro Tiago C. |last3=de Araújo |first3=João M. |last4=Corso |first4=Gilberto |date=2020-05-27 |title=Full-waveform inversion based on Kaniadakis statistics |url=https://link.aps.org/doi/10.1103/PhysRevE.101.053311 |journal=Physical Review E |language=en |volume=101 |issue=5 |pages=053311 |doi=10.1103/PhysRevE.101.053311 |pmid=32575242 |bibcode=2020PhRvE.101e3311D |s2cid=219746493 |issn=2470-0045}} 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:{{Cite journal |last=Kaniadakis |first=G. |date=2021-01-01 |title=New power-law tailed distributions emerging in κ-statistics (a) |url=https://iopscience.iop.org/article/10.1209/0295-5075/133/10002 |journal=Europhysics Letters |volume=133 |issue=1 |pages=10002 |doi=10.1209/0295-5075/133/10002 |arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 |issn=0295-5075}}

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

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

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

is the normalization constant.

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

= Cumulative distribution function =

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

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

where

$\textrm{erf}_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma\Big( \frac{1}{2\kappa} + \frac{1}{4} \Big)}{ \Gamma\Big( \frac{1}{2\kappa} - \frac{1}{4} \Big) } \int_0^x \exp_\kappa(-t^2 )
dt$

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

\textrm{erf}(x)

\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:

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

= Kurtosis =

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

: $\operatorname{Kurt}[X] = \operatorname{E}\left[\frac{X^4}{\sigma_\kappa^4}\right]$

which can be written as

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

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

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

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

κ-Error function

{{Infobox mathematical function

| name = κ-Error function

| image = File:Kappa error function pdf.png

| imagesize = 400px

| imagealt = Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.

| caption = Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.

| general_definition = $\operatorname{erf}_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma\Big( \frac{1}{2\kappa} + \frac{1}{4} \Big)}{ \Gamma\Big( \frac{1}{2\kappa} - \frac{1}{4} \Big) } \int_0^x \exp_\kappa(-t^2 )
dt$

| fields_of_application = Probability, thermodynamics

| domain = $\mathbb{C}$

| range = $\left( -1,1 \right)$

|root= $0$ |derivative= $\frac{d}{dx}\operatorname{erf}_\kappa(x) = \left( 2+ \kappa \right) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma \left( \frac{1}{2 \kappa} + \frac{1}{4} \right)}{ \Gamma \left( \frac{1}{2 \kappa} - \frac{1}{4} \right) } \exp_\kappa(-x^2 )$ }}

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

: $\operatorname{erf}_\kappa(x) = \Big( 2+ \kappa \Big) \sqrt{ \frac{2 \kappa}{\pi} } \frac{\Gamma\Big( \frac{1}{2\kappa} + \frac{1}{4} \Big)}{ \Gamma\Big( \frac{1}{2\kappa} - \frac{1}{4} \Big) } \int_0^x \exp_\kappa(-t^2 )
dt$

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

For a random variable {{mvar|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.{{Cite journal |last1=Moretto |first1=Enrico |last2=Pasquali |first2=Sara |last3=Trivellato |first3=Barbara |date=2017 |title=A non-Gaussian option pricing model based on Kaniadakis exponential deformation |url=http://link.springer.com/10.1140/epjb/e2017-80112-x |journal=The European Physical Journal B |language=en |volume=90 |issue=10 |pages=179 |doi=10.1140/epjb/e2017-80112-x |bibcode=2017EPJB...90..179M |s2cid=254116243 |issn=1434-6028}}
In inverse problems, Error laws in extreme statistics are robustly represented by κ-Gaussian distributions.{{Cite journal |last1=Wada |first1=Tatsuaki |last2=Suyari |first2=Hiroki |date=2006 |title=κ-generalization of Gauss' law of error |url=https://linkinghub.elsevier.com/retrieve/pii/S0375960105013630 |journal=Physics Letters A |language=en |volume=348 |issue=3–6 |pages=89–93 |doi=10.1016/j.physleta.2005.08.086|arxiv=cond-mat/0505313 |bibcode=2006PhLA..348...89W |s2cid=119003351 }}{{Cite journal |last1=da Silva |first1=Sérgio Luiz E.F. |last2=Silva |first2=R. |last3=dos Santos Lima |first3=Gustavo Z. |last4=de Araújo |first4=João M. |last5=Corso |first5=Gilberto |date=2022 |title=An outlier-resistant κ -generalized approach for robust physical parameter estimation |url=https://linkinghub.elsevier.com/retrieve/pii/S0378437122003855 |journal=Physica A: Statistical Mechanics and Its Applications |language=en |volume=600 |pages=127554 |doi=10.1016/j.physa.2022.127554|arxiv=2111.09921 |bibcode=2022PhyA..60027554D |s2cid=248803855 }}
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.{{Cite journal |last1=Carvalho |first1=J. C. |last2=Silva |first2=R. |last3=do Nascimento jr. |first3=J. D. |last4=Soares |first4=B. B. |last5=De Medeiros |first5=J. R. |date=2010-09-01 |title=Observational measurement of open stellar clusters: A test of Kaniadakis and Tsallis statistics |url=https://iopscience.iop.org/article/10.1209/0295-5075/91/69002 |journal=EPL (Europhysics Letters) |volume=91 |issue=6 |pages=69002 |doi=10.1209/0295-5075/91/69002 |bibcode=2010EL.....9169002C |s2cid=120902898 |issn=0295-5075}}{{Cite journal |last1=Carvalho |first1=J. C. |last2=Silva |first2=R. |last3=do Nascimento jr. |first3=J. D. |last4=De Medeiros |first4=J. R. |date=2008 |title=Power law statistics and stellar rotational velocities in the Pleiades |url=https://iopscience.iop.org/article/10.1209/0295-5075/84/59001 |journal=EPL (Europhysics Letters) |volume=84 |issue=5 |pages=59001 |doi=10.1209/0295-5075/84/59001 |arxiv=0903.0836 |bibcode=2008EL.....8459001C |s2cid=7123391 |issn=0295-5075}}
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.{{Cite journal |last1=Guedes |first1=Guilherme |last2=Gonçalves |first2=Alessandro C. |last3=Palma |first3=Daniel A.P. |date=2017 |title=The Doppler Broadening Function using the Kaniadakis distribution |url=https://linkinghub.elsevier.com/retrieve/pii/S030645491730155X |journal=Annals of Nuclear Energy |language=en |volume=110 |pages=453–458 |doi=10.1016/j.anucene.2017.06.057}}{{Cite journal |last1=de Abreu |first1=Willian V. |last2=Gonçalves |first2=Alessandro C. |last3=Martinez |first3=Aquilino S. |date=2019 |title=Analytical solution for the Doppler broadening function using the Kaniadakis distribution |url=https://linkinghub.elsevier.com/retrieve/pii/S0306454918306224 |journal=Annals of Nuclear Energy |language=en |volume=126 |pages=262–268 |doi=10.1016/j.anucene.2018.11.023|s2cid=125724227 }}
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{{Cite journal |last1=Gougam |first1=Leila Ait |last2=Tribeche |first2=Mouloud |date=2016 |title=Electron-acoustic waves in a plasma with a κ -deformed Kaniadakis electron distribution |url=http://aip.scitation.org/doi/10.1063/1.4939477 |journal=Physics of Plasmas |language=en |volume=23 |issue=1 |pages=014501 |doi=10.1063/1.4939477 |bibcode=2016PhPl...23a4501G |issn=1070-664X}} and the dispersion of Langmuir waves.{{Cite journal |last1=Chen |first1=H. |last2=Zhang |first2=S. X. |last3=Liu |first3=S. Q. |date=2017 |title=The longitudinal plasmas modes of κ -deformed Kaniadakis distributed plasmas |url=http://aip.scitation.org/doi/10.1063/1.4976992 |journal=Physics of Plasmas |language=en |volume=24 |issue=2 |pages=022125 |doi=10.1063/1.4976992 |bibcode=2017PhPl...24b2125C |issn=1070-664X}}

References

External links

[https://arxiv.org/search/?query=kaniadakis+statistics&searchtype=all&abstracts=show&order=-announced_date_first&size=200 Kaniadakis Statistics on arXiv.org]

Category:Probability distributions

Category:Mathematical and quantitative methods (economics)