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Kaniadakis exponential distribution

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The Kaniadakis exponential distribution (or κ-exponential distribution) is a probability distribution arising from the maximization of the Kaniadakis entropy under appropriate constraints. It is one example of a Kaniadakis distribution. The κ-exponential is a generalization of the exponential distribution in the same way that Kaniadakis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.[1] The κ-exponential distribution of Type I is a particular case of the κ-Gamma distribution, whilst the κ-exponential distribution of Type II is a particular case of the κ-Weibull distribution.

Type I

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Probability density function

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κ-exponential distribution of type I
Probability density function
Cumulative distribution function
Parameters shape (real)
rate (real)
Support
PDF
CDF
Mean
Variance
Skewness
Excess kurtosis
Method of moments

The Kaniadakis κ-exponential distribution of Type I is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics which exhibit power-law tails. This distribution has the following probability density function:[2]

valid for , where is the entropic index associated with the Kaniadakis entropy and is known as rate parameter. The exponential distribution is recovered as

Cumulative distribution function

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The cumulative distribution function of κ-exponential distribution of Type I is given by

for . The cumulative exponential distribution is recovered in the classical limit .

Properties

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Moments, expectation value and variance

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The κ-exponential distribution of type I has moment of order given by[2]

where is finite if .

The expectation is defined as:

and the variance is:

Kurtosis

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The kurtosis of the κ-exponential distribution of type I may be computed thought:

Thus, the kurtosis of the κ-exponential distribution of type I distribution is given by:

or

The kurtosis of the ordinary exponential distribution is recovered in the limit .

Skewness

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The skewness of the κ-exponential distribution of type I may be computed thought:

Thus, the skewness of the κ-exponential distribution of type I distribution is given by:

The kurtosis of the ordinary exponential distribution is recovered in the limit .

Type II

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Probability density function

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κ-exponential distribution of type II
Probability density function
Cumulative distribution function
Parameters shape (real)
rate (real)
Support
PDF
CDF
Quantile
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Method of moments

The Kaniadakis κ-exponential distribution of Type II also is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics which exhibit power-law tails, but with different constraints. This distribution is a particular case of the Kaniadakis κ-Weibull distribution with is:[2]

valid for , where is the entropic index associated with the Kaniadakis entropy and is known as rate parameter.

The exponential distribution is recovered as

Cumulative distribution function

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The cumulative distribution function of κ-exponential distribution of Type II is given by

for . The cumulative exponential distribution is recovered in the classical limit .

Properties

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Moments, expectation value and variance

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The κ-exponential distribution of type II has moment of order given by[2]

The expectation value and the variance are:

The mode is given by:

Kurtosis

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The kurtosis of the κ-exponential distribution of type II may be computed thought:

Thus, the kurtosis of the κ-exponential distribution of type II distribution is given by:

or

Skewness

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The skewness of the κ-exponential distribution of type II may be computed thought:

Thus, the skewness of the κ-exponential distribution of type II distribution is given by:

or

The skewness of the ordinary exponential distribution is recovered in the limit .

Quantiles

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The quantiles are given by the following expression

with , in which the median is the case :

Lorenz curve

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The Lorenz curve associated with the κ-exponential distribution of type II is given by:[2]

The Gini coefficient is

Asymptotic behavior

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The κ-exponential distribution of type II behaves asymptotically as follows:[2]

Applications

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

See also

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References

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  1. ^ 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.
  2. ^ a b c d e f 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.
  3. ^ Oreste, Pierpaolo; Spagnoli, Giovanni (2018-04-03). "Statistical analysis of some main geomechanical formulations evaluated with the Kaniadakis exponential law". Geomechanics and Geoengineering. 13 (2): 139–145. doi:10.1080/17486025.2017.1373201. ISSN 1748-6025. S2CID 133860553.
  4. ^ Ourabah, Kamel; Tribeche, Mouloud (2014). "Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics". Physical Review E. 89 (6): 062130. Bibcode:2014PhRvE..89f2130O. doi:10.1103/PhysRevE.89.062130. ISSN 1539-3755. PMID 25019747.
  5. ^ 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.
  6. ^ Macedo-Filho, A.; Moreira, D.A.; Silva, R.; da Silva, Luciano R. (2013). "Maximum entropy principle for Kaniadakis statistics and networks". Physics Letters A. 377 (12): 842–846. Bibcode:2013PhLA..377..842M. doi:10.1016/j.physleta.2013.01.032.
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