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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.

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]

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 ${\textrm {erf}}(x)$ 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 {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

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

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

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

or

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

κ-Error function

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

\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 ${\sqrt {\beta }}$ , 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.^[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]

References

^ Moretto, Enrico; Pasquali, Sara; Trivellato, Barbara (2017). "A non-Gaussian option pricing model based on Kaniadakis exponential deformation". The European Physical Journal B. 90 (10): 179. Bibcode:2017EPJB...90..179M. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028. S2CID 254116243.
^ ^a ^b da Silva, Sérgio Luiz E. F.; Carvalho, Pedro Tiago C.; de Araújo, João M.; Corso, Gilberto (2020-05-27). "Full-waveform inversion based on Kaniadakis statistics". Physical Review E. 101 (5): 053311. Bibcode:2020PhRvE.101e3311D. doi:10.1103/PhysRevE.101.053311. ISSN 2470-0045. PMID 32575242. S2CID 219746493.
^ ^a ^b ^c Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.
^ Moretto, Enrico; Pasquali, Sara; Trivellato, Barbara (2017). "A non-Gaussian option pricing model based on Kaniadakis exponential deformation". The European Physical Journal B. 90 (10): 179. Bibcode:2017EPJB...90..179M. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028. S2CID 254116243.
^ Wada, Tatsuaki; Suyari, Hiroki (2006). "κ-generalization of Gauss' law of error". Physics Letters A. 348 (3–6): 89–93. arXiv:cond-mat/0505313. Bibcode:2006PhLA..348...89W. doi:10.1016/j.physleta.2005.08.086. S2CID 119003351.
^ da Silva, Sérgio Luiz E.F.; Silva, R.; dos Santos Lima, Gustavo Z.; de Araújo, João M.; Corso, Gilberto (2022). "An outlier-resistant κ -generalized approach for robust physical parameter estimation". Physica A: Statistical Mechanics and Its Applications. 600: 127554. arXiv:2111.09921. Bibcode:2022PhyA..60027554D. doi:10.1016/j.physa.2022.127554. S2CID 248803855.
^ Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; Soares, B. B.; De Medeiros, J. R. (2010-09-01). "Observational measurement of open stellar clusters: A test of Kaniadakis and Tsallis statistics". EPL (Europhysics Letters). 91 (6): 69002. Bibcode:2010EL.....9169002C. doi:10.1209/0295-5075/91/69002. ISSN 0295-5075. S2CID 120902898.
^ Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; De Medeiros, J. R. (2008). "Power law statistics and stellar rotational velocities in the Pleiades". EPL (Europhysics Letters). 84 (5): 59001. arXiv:0903.0836. Bibcode:2008EL.....8459001C. doi:10.1209/0295-5075/84/59001. ISSN 0295-5075. S2CID 7123391.
^ Guedes, Guilherme; Gonçalves, Alessandro C.; Palma, Daniel A.P. (2017). "The Doppler Broadening Function using the Kaniadakis distribution". Annals of Nuclear Energy. 110: 453–458. doi:10.1016/j.anucene.2017.06.057.
^ de Abreu, Willian V.; Gonçalves, Alessandro C.; Martinez, Aquilino S. (2019). "Analytical solution for the Doppler broadening function using the Kaniadakis distribution". Annals of Nuclear Energy. 126: 262–268. doi:10.1016/j.anucene.2018.11.023. S2CID 125724227.
^ Gougam, Leila Ait; Tribeche, Mouloud (2016). "Electron-acoustic waves in a plasma with a κ -deformed Kaniadakis electron distribution". Physics of Plasmas. 23 (1): 014501. Bibcode:2016PhPl...23a4501G. doi:10.1063/1.4939477. ISSN 1070-664X.
^ Chen, H.; Zhang, S. X.; Liu, S. Q. (2017). "The longitudinal plasmas modes of κ -deformed Kaniadakis distributed plasmas". Physics of Plasmas. 24 (2): 022125. Bibcode:2017PhPl...24b2125C. doi:10.1063/1.4976992. ISSN 1070-664X.

External links

Kaniadakis Statistics on arXiv.org

[1] Moretto, Enrico; Pasquali, Sara; Trivellato, Barbara (2017). "A non-Gaussian option pricing model based on Kaniadakis exponential deformation". The European Physical Journal B. 90 (10): 179. Bibcode:2017EPJB...90..179M. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028. S2CID 254116243.

[:1-2] Silva, Sérgio Luiz E. F.; Carvalho, Pedro Tiago C.; de Araújo, João M.; Corso, Gilberto (2020-05-27). "Full-waveform inversion based on Kaniadakis statistics". Physical Review E. 101 (5): 053311. Bibcode:2020PhRvE.101e3311D. doi:10.1103/PhysRevE.101.053311. ISSN 2470-0045. PMID 32575242. S2CID 219746493.

[:0-3] Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.

[4] Moretto, Enrico; Pasquali, Sara; Trivellato, Barbara (2017). "A non-Gaussian option pricing model based on Kaniadakis exponential deformation". The European Physical Journal B. 90 (10): 179. Bibcode:2017EPJB...90..179M. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028. S2CID 254116243.

[5] Wada, Tatsuaki; Suyari, Hiroki (2006). "κ-generalization of Gauss' law of error". Physics Letters A. 348 (3–6): 89–93. arXiv:cond-mat/0505313. Bibcode:2006PhLA..348...89W. doi:10.1016/j.physleta.2005.08.086. S2CID 119003351.

[6] Silva, Sérgio Luiz E.F.; Silva, R.; dos Santos Lima, Gustavo Z.; de Araújo, João M.; Corso, Gilberto (2022). "An outlier-resistant κ -generalized approach for robust physical parameter estimation". Physica A: Statistical Mechanics and Its Applications. 600: 127554. arXiv:2111.09921. Bibcode:2022PhyA..60027554D. doi:10.1016/j.physa.2022.127554. S2CID 248803855.

[7] Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; Soares, B. B.; De Medeiros, J. R. (2010-09-01). "Observational measurement of open stellar clusters: A test of Kaniadakis and Tsallis statistics". EPL (Europhysics Letters). 91 (6): 69002. Bibcode:2010EL.....9169002C. doi:10.1209/0295-5075/91/69002. ISSN 0295-5075. S2CID 120902898.

[8] Carvalho, J. C.; Silva, R.; do Nascimento jr., J. D.; De Medeiros, J. R. (2008). "Power law statistics and stellar rotational velocities in the Pleiades". EPL (Europhysics Letters). 84 (5): 59001. arXiv:0903.0836. Bibcode:2008EL.....8459001C. doi:10.1209/0295-5075/84/59001. ISSN 0295-5075. S2CID 7123391.

[9] Guedes, Guilherme; Gonçalves, Alessandro C.; Palma, Daniel A.P. (2017). "The Doppler Broadening Function using the Kaniadakis distribution". Annals of Nuclear Energy. 110: 453–458. doi:10.1016/j.anucene.2017.06.057.

[10] Abreu, Willian V.; Gonçalves, Alessandro C.; Martinez, Aquilino S. (2019). "Analytical solution for the Doppler broadening function using the Kaniadakis distribution". Annals of Nuclear Energy. 126: 262–268. doi:10.1016/j.anucene.2018.11.023. S2CID 125724227.

[11] Gougam, Leila Ait; Tribeche, Mouloud (2016). "Electron-acoustic waves in a plasma with a κ -deformed Kaniadakis electron distribution". Physics of Plasmas. 23 (1): 014501. Bibcode:2016PhPl...23a4501G. doi:10.1063/1.4939477. ISSN 1070-664X.

[12] Chen, H.; Zhang, S. X.; Liu, S. Q. (2017). "The longitudinal plasmas modes of κ -deformed Kaniadakis distributed plasmas". Physics of Plasmas. 24 (2): 022125. Bibcode:2017PhPl...24b2125C. doi:10.1063/1.4976992. ISSN 1070-664X.

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