κ-Gaussian distribution
Probability density function
Kaniadakis Gaussian Distribution Type II pdf.png
Cumulative distribution function
Kaniadakis Gaussian Distribution Type II cdf.png
Parameters shape (real)
rate (real)
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. 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]

Definitions

Probability density function

The general form of the centered Kaniadakis κ-Gaussian probability density function is:[3]

where is the entropic index associated with the Kaniadakis entropy, is the scale parameter, and

is the normalization constant.

The standard Normal distribution is recovered in the limit

Cumulative distribution function

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

where

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

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 and is given by:

Kurtosis

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

which can be written as

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

or

κ-Error function

κ-Error function
Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.
Plot of the κ-error function for typical κ-values. The case κ=0 corresponds to the ordinary error function.
General information
General definition
Fields of applicationProbability, thermodynamics
Domain, Codomain and Image
Domain
Image
Specific features
Root
Derivative

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

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 , κ-Error function means the probability that X falls in the interval .

Applications

The κ-Gaussian distribution has been applied in several areas, such as:

See also

References

  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. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028.
  2. ^ 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. doi:10.1103/PhysRevE.101.053311. ISSN 2470-0045.
  3. ^ a b c Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075.
  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. doi:10.1140/epjb/e2017-80112-x. ISSN 1434-6028.
  5. ^ Wada, Tatsuaki; Suyari, Hiroki (2006). "κ-generalization of Gauss' law of error". Physics Letters A. 348 (3–6): 89–93. doi:10.1016/j.physleta.2005.08.086.
  6. ^ 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. doi:10.1016/j.physa.2022.127554.
  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. doi:10.1209/0295-5075/91/69002. ISSN 0295-5075.
  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. doi:10.1209/0295-5075/84/59001. ISSN 0295-5075.
  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. ^ 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.
  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. 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. doi:10.1063/1.4976992. ISSN 1070-664X.