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In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics. There are several families of Kaniadakis distributions related to different constraints used in the maximization of the Kaniadakis entropy, such as the κ-Exponential distribution, κ-Gaussian distribution, Kaniadakis κ-Gamma distribution and κ-Weibull distribution. The κ-distributions have been applied for modeling a vast phenomenology of experimental statistical distributions in natural or artificial complex systems, such as, in epidemiology, quantum statistics, in astrophysics and cosmology, in geophysics, in economy, in machine learning.

The κ-distributions are written as function of the κ-deformed exponential, taking the form

${\displaystyle f_{i}=\exp _{\kappa }(-\beta E_{i}+\beta \mu )}$

enables the power-law description of complex systems following the consistent κ-generalized statistical theory., where ${\displaystyle \exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x)^{1/\kappa }}$ is the Kaniadakis κ-exponential function.

The κ-distribution becomes the common Boltzmann distribution at low energies, while it has a power-law tail at high energies, the feature of high interest of many researchers.

List of κ-statistical distributions

Supported on the whole real line

• The Kaniadakis Gaussian distribution, also called the κ-Gaussian distribution. The normal distribution is a particular case when ${\displaystyle \kappa \rightarrow 0.}$
• The Kaniadakis double exponential distribution, as known as Kaniadakis κ-double exponential distribution or κ-Laplace distribution. The Laplace distribution is a particular case when ${\displaystyle \kappa \rightarrow 0.}$

Supported on semi-infinite intervals, usually

• The Kaniadakis Exponential distribution, also called the κ-Exponential distribution. The exponential distribution is a particular case when ${\displaystyle \kappa \rightarrow 0.}$
• The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter (${\displaystyle \kappa ,\alpha ,\beta ,\nu }$) deformation of the generalized Gamma distribution.
  • The κ-Gamma distribution becomes a ...
    • κ-Exponential distribution of Type I when ${\displaystyle \alpha =\nu =1}$.
    • κ-Erlang distribution when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =n=}$ positive integer.
    • κ-Half-Normal distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1/2}$.
    • Generalized Gamma distribution, when ${\displaystyle \alpha =1}$;
  • In the limit ${\displaystyle \kappa \rightarrow 0}$, the κ-Gamma distribution becomes a ...
    • Erlang distribution, when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =n=}$ positive integer;
    • Chi-Squared distribution, when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =}$ half integer;
    • Nakagami distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu >0}$;
    • Rayleigh distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1}$;
    • Chi distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =}$ half integer;
    • Maxwell distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =3/2}$;
    • Half-Normal distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1/2}$;
    • Weibull distribution, when ${\displaystyle \alpha >0}$ and ${\displaystyle \nu =1}$;
    • Stretched Exponential distribution, when ${\displaystyle \alpha >0}$ and ${\displaystyle \nu =1/\alpha }$;

Common Kaniadakis distributions

κ-Exponential distribution

κ-Gaussian distribution

κ-Gamma distribution

κ-Weibull distribution

κ-Logistic distribution

κ-Erlang distribution

κ-Distribution Type IV

κ-Distribution Type IV

Probability density functionPlot of the κ-Distribution Type IV for typical κ-values, and ${\displaystyle \alpha =\beta =1}$.
Cumulative distribution function
Parameters	${\displaystyle 0\leq \kappa <1}$ ${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ rate (real)
Support	${\displaystyle x\in [0,+\infty )}$
PDF	${\displaystyle {\frac {\alpha }{\kappa }}(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha }}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$
CDF	${\displaystyle (2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$
Method of moments	${\displaystyle {\frac {(2\kappa \beta )^{-m/\alpha }}{1+\kappa {\frac {m}{2\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{\alpha }}{\Big )}\Gamma {\Big (}1-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}}$

The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.

The κ-Distribution Type IV distribution has the following probability density function:

${\displaystyle f_{_{\kappa }}(x)={\frac {\alpha }{\kappa }}(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha }}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$

valid for ${\displaystyle x\geq 0}$, where ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index associated with the Kaniadakis entropy, ${\displaystyle \beta >0}$ is the scale parameter, and ${\displaystyle \alpha >0}$ is the shape parameter.

The cumulative distribution function of κ-Distribution Type IV assumes the form:

${\displaystyle F_{\kappa }(x)=(2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$

The κ-Distribution Type IV does not admit a classical version, since the probability function and its cumulative reduces to zero in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

Its moment of order ${\displaystyle m}$ given by

${\displaystyle \operatorname {E} ={\frac {(2\kappa \beta )^{-m/\alpha }}{1+\kappa {\frac {m}{2\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{\alpha }}{\Big )}\Gamma {\Big (}1-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}}$

The moment of order ${\displaystyle m}$ of the κ-Distribution Type IV is finite for ${\displaystyle m<2\alpha }$.

See also

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

References

1. ^ 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.
2. Kaniadakis, Giorgio; Baldi, Mauro M.; Deisboeck, Thomas S.; Grisolia, Giulia; Hristopulos, Dionissios T.; Scarfone, Antonio M.; Sparavigna, Amelia; Wada, Tatsuaki; Lucia, Umberto (2020). "The κ-statistics approach to epidemiology". Scientific Reports. 10 (1): 19949. arXiv:2012.00629. Bibcode:2020NatSR..1019949K. doi:10.1038/s41598-020-76673-3. ISSN 2045-2322. PMC 7673996. PMID 33203913.
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9. Hristopulos, Dionissios T.; Petrakis, Manolis P.; Kaniadakis, Giorgio (2014-05-28). "Finite-size effects on return interval distributions for weakest-link-scaling systems". Physical Review E. 89 (5): 052142. arXiv:1308.1881. Bibcode:2014PhRvE..89e2142H. doi:10.1103/PhysRevE.89.052142. ISSN 1539-3755. PMID 25353774. S2CID 22310350.
10. da Silva, Sérgio Luiz E.F. (2021). "κ -generalised Gutenberg–Richter law and the self-similarity of earthquakes". Chaos, Solitons & Fractals. 143: 110622. Bibcode:2021CSF...14310622D. doi:10.1016/j.chaos.2020.110622. S2CID 234063959.
11. 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.
12. Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio; Landini, Simone (2016). "κ-generalized models of income and wealth distributions: A survey". The European Physical Journal Special Topics. 225 (10): 1959–1984. arXiv:1610.08676. Bibcode:2016EPJST.225.1959C. doi:10.1140/epjst/e2016-60014-2. ISSN 1951-6355. S2CID 125503224.
13. Clementi, Fabio; Gallegati, Mauro; Kaniadakis, Giorgio (2012). "A new model of income distribution: the κ-generalized distribution". Journal of Economics. 105 (1): 63–91. doi:10.1007/s00712-011-0221-0. hdl:11393/73598. ISSN 0931-8658. S2CID 155080665.
14. Trivellato, Barbara (2013-09-02). "Deformed Exponentials and Applications to Finance" (PDF). Entropy. 15 (12): 3471–3489. Bibcode:2013Entrp..15.3471T. doi:10.3390/e15093471. ISSN 1099-4300.
15. Passos, Leandro Aparecido; Cleison Santana, Marcos; Moreira, Thierry; Papa, Joao Paulo (2019). "κ-Entropy Based Restricted Boltzmann Machines". 2019 International Joint Conference on Neural Networks (IJCNN). Budapest, Hungary: IEEE. pp. 1–8. doi:10.1109/IJCNN.2019.8851714. ISBN 978-1-7281-1985-4. S2CID 203605811.
16. Kaniadakis, Giorgio (2013-09-25). "Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions". Entropy. 15 (12): 3983–4010. arXiv:1309.6536. Bibcode:2013Entrp..15.3983K. doi:10.3390/e15103983. ISSN 1099-4300.
17. Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics". Physica A: Statistical Mechanics and Its Applications. 296 (3–4): 405–425. arXiv:cond-mat/0103467. Bibcode:2001PhyA..296..405K. doi:10.1016/S0378-4371(01)00184-4. S2CID 44275064.
18. da Silva, Sérgio Luiz E. F.; dos Santos Lima, Gustavo Z.; Volpe, Ernani V.; de Araújo, João M.; Corso, Gilberto (2021). "Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics". The European Physical Journal Plus. 136 (5): 518. Bibcode:2021EPJP..136..518D. doi:10.1140/epjp/s13360-021-01521-w. ISSN 2190-5444. S2CID 236575441.

External links

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org

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• Probability distributions