Kaniadakis Gaussian distribution information

κ-Gaussian distribution
	Probability density function
	Cumulative distribution function
Parameters	shape (real) ; rate (real)
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Excess kurtosis

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]

^ 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.
^ 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.
^ Cite error: The named reference :0 was invoked but never defined (see the help page).

[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] Cite error: The named reference :0 was invoked but never defined (see the help page).

Kaniadakis Gaussian distribution information

and 14 Related for: Kaniadakis Gaussian distribution information

Kaniadakis Gaussian distribution

Normal distribution

Kaniadakis Gamma distribution

Kaniadakis distribution

Multivariate normal distribution

Kaniadakis Erlang distribution

Kaniadakis logistic distribution

Kaniadakis Weibull distribution

Kaniadakis exponential distribution

Kaniadakis statistics

Giorgio Kaniadakis

Elliptical distribution

Multivariate Laplace distribution

Tsallis statistics

κ-Gaussian distribution
Probability density function
Cumulative distribution function
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 )}\$
Mean	$0$
Median	$0$
Mode	$0$
Variance	$\sigma _{\kappa }^{2}={\frac {1}{\beta }}{\frac {2+\kappa }{2-\kappa }}{\frac {4\kappa }{4-9\kappa ^{2}}}\left[{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{4}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{4}}{\Big )}}}\right]^{2}$
Skewness	$0$
Excess kurtosis	$3\left[{\frac {{\sqrt {\pi }}Z_{\kappa }}{2\beta ^{2/3}\sigma _{\kappa }^{4}}}{\frac {(2\kappa )^{-5/2}}{1+{\frac {5}{2}}\kappa }}{\frac {\Gamma \left({\frac {1}{2\kappa }}-{\frac {5}{4}}\right)}{\Gamma \left({\frac {1}{2\kappa }}+{\frac {5}{4}}\right)}}-1\right]$