Kaniadakis exponential distribution

Short description: Probability distribution

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.

## Contents

* 1 Type I
* 1.1 Probability density function
* 1.2 Cumulative distribution function
* 1.3 Properties
* 1.3.1 Moments, expectation value and variance
* 1.3.2 Kurtosis
* 1.3.3 Skewness
* 2 Type II
* 2.1 Probability density function
* 2.2 Cumulative distribution function
* 2.3 Properties
* 2.3.1 Moments, expectation value and variance
* 2.3.2 Kurtosis
* 2.3.3 Skewness
* 2.3.4 Quantiles
* 2.3.5 Lorenz curve
* 2.3.6 Asymptotic behavior
* 3 Applications
* 4 See also
* 5 References
* 6 External links

## Type I

### Probability density function

κ-exponential distribution of type I

Probability density function

Cumulative distribution function

Parameters|  0<κ<1 shape (real)
β>0 rate (real)
Support|  x∈[0,∞)
PDF|  (1−κ2)βexpκ(−βx)
CDF|  1−(1+κ2β2x2+κ2βx)expk(−βx)
Mean|  1β1−κ21−4κ2
Variance|  σκ2=1β22(1−4κ2)2−(1−κ2)2(1−9κ2)(1−4κ2)2(1−9κ2)
Skewness|  2(1−κ2)(144κ8+23κ6+27κ4−6κ2+1)β3σκ3(4κ2−1)3(144κ4−25κ2+1)
Kurtosis|  9(1200κ14−6123κ12+562κ10+1539κ8−544κ6+143κ4−18κ2+1)β4σκ4(1−κ2)−1(1−4κ2)4(3600κ8−4369κ6+819κ4−51κ2+1)−3

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]

fκ(x)=(1−κ2)βexpκ(−βx)

valid for x≥0, where 0≤|κ|<1 is the entropic index associated with the Kaniadakis entropy and β>0 is known as rate parameter. The exponential distribution is recovered as κ->0.

### Cumulative distribution function

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

Fκ(x)=1−(1+κ2β2x2+κ2βx)expk(−βx)

for x≥0. The cumulative exponential distribution is recovered in the classical limit κ->0.

### Properties

#### Moments, expectation value and variance

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

E⁡[Xm]=1−κ2∏n=0m+1[1−(2n−m−1)κ]m!βm

where fκ(x) is finite if 0<m+1<1/κ.

The expectation is defined as:

E⁡[X]=1β1−κ21−4κ2

and the variance is:

Var⁡[X]=σκ2=1β22(1−4κ2)2−(1−κ2)2(1−9κ2)(1−4κ2)2(1−9κ2)

#### Kurtosis

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

Kurt⁡[X]=E⁡[[X−1β1−κ21−4κ2]4σκ4]

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

> Kurt⁡[X]=9(1−κ2)(1200κ14−6123κ12+562κ10+1539κ8−544κ6+143κ4−18κ2+1)β4σκ4(1−4κ2)4(3600κ8−4369κ6+819κ4−51κ2+1)for0≤κ<1/5

> Kurt⁡[X]=9(9κ2−1)2(κ2−1)(1200κ14−6123κ12+562κ10+1539κ8−544κ6+143κ4−18κ2+1)β2(1−4κ2)2(9κ6+13κ4−5κ2+1)(3600κ8−4369κ6+819κ4−51κ2+1)for0≤κ<1/5

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

κ->0

#### Skewness

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

Skew⁡[X]=E⁡[[X−1β1−κ21−4κ2]3σκ3]

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

> Shew⁡[X]=2(1−κ2)(144κ8+23κ6+27κ4−6κ2+1)β3σκ3(4κ2−1)3(144κ4−25κ2+1)for0≤κ<1/4

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

κ->0

## Type II

### Probability density function

κ-exponential distribution of type II

Probability density function

Cumulative distribution function

Parameters|  0≤κ<1 shape (real)
β>0 rate (real)
Support|  x∈[0,∞)
PDF|  β1+κ2β2x2expκ(−βx)
CDF|  1−expk(−βx)
Quantile|  β−1lnκ(11−Fκ),0≤Fκ≤1
Mean|  1β11−κ2
Median|  β−1lnκ(2)
Mode|  1κβ2(1−κ2)
Variance|  σκ2=1β21+2κ4(1−4κ2)(1−κ2)2
Skewness|  2(15κ6+6κ4+2κ2+1)(1−9κ2)(2κ4+1)1−4κ21+2κ4
Kurtosis|  3(72κ10−360κ8−44κ6−32κ4+7κ2−3)(4κ2−1)−1(2κ4+1)2(144κ4−25κ2+1)

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 α=1 is:[2]

fκ(x)=β1+κ2β2x2expκ(−βx)

valid for x≥0, where 0≤|κ|<1 is the entropic index associated with the Kaniadakis entropy and β>0 is known as rate parameter.

The exponential distribution is recovered as κ->0.

### Cumulative distribution function

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

Fκ(x)=1−expk(−βx)

for x≥0. The cumulative exponential distribution is recovered in the classical limit κ->0.

### Properties

#### Moments, expectation value and variance

The κ-exponential distribution of type II has moment of order m<1/κ given by[2]

E⁡[Xm]=β−mm!∏n=0m[1−(2n−m)κ]

The expectation value and the variance are:

E⁡[X]=1β11−κ2

Var⁡[X]=σκ2=1β21+2κ4(1−4κ2)(1−κ2)2

The mode is given by:

xmode=1κβ2(1−κ2)

#### Kurtosis

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

Kurt⁡[X]=E⁡[(X−1β11−κ2σκ)4]

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

Kurt⁡[X]=3(72κ10−360κ8−44κ6−32κ4+7κ2−3)β4σκ4(κ2−1)4(576κ6−244κ4+29κ2−1) for 0≤κ<1/4

Kurt⁡[X]=3(72κ10−360κ8−44κ6−32κ4+7κ2−3)(4κ2−1)−1(2κ4+1)2(144κ4−25κ2+1) for 0≤κ<1/4

#### Skewness

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

Skew⁡[X]=E⁡[[X−1β11−κ2]3σκ3]

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

> Skew⁡[X]=−2(15κ6+6κ4+2κ2+1)β3σκ3(κ2−1)3(36κ4−13κ2+1)for0≤κ<1/3

> Skew⁡[X]=2(15κ6+6κ4+2κ2+1)(1−9κ2)(2κ4+1)1−4κ21+2κ4for0≤κ<1/3

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

κ->0

#### Quantiles

The quantiles are given by the following expression

> xquantile(Fκ)=β−1lnκ(11−Fκ)

with

0≤Fκ≤1

, in which the median is the case :

> xmedian(Fκ)=β−1lnκ(2)

#### Lorenz curve

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

ℒκ(Fκ)=1+1−κ2κ(1−Fκ)1+κ−1+κ2κ(1−Fκ)1−κ

The Gini coefficient is

> Gκ=2+κ24−κ2

#### Asymptotic behavior

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

limx->+∞fκ(x)∼κ−1(2κβ)−1/κx(−1−κ)/κ
limx->0+fκ(x)=β

## Applications

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

* In geomechanics, for analyzing the properties of rock masses;[3]
* In quantum theory, in physical analysis using Planck's radiation law;[4]
* In inverse problems, the κ-exponential distribution has been used to formulate a robust approach;[5]
* In Network theory.[6]

## See also

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

## References

1. ↑ Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics" (in en). Physica A: Statistical Mechanics and Its Applications 296 (3–4): 405–425. doi:10.1016/S0378-4371(01)00184-4. Bibcode: 2001PhyA..296..405K. https://linkinghub.elsevier.com/retrieve/pii/S0378437101001844.
2. ↑ 2.0 2.1 2.2 2.3 2.4 2.5 Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters 133 (1). doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. Bibcode: 2021EL....13310002K. https://iopscience.iop.org/article/10.1209/0295-5075/133/10002.
3. ↑ Oreste, Pierpaolo; Spagnoli, Giovanni (2018-04-03). "Statistical analysis of some main geomechanical formulations evaluated with the Kaniadakis exponential law" (in en). Geomechanics and Geoengineering 13 (2): 139–145. doi:10.1080/17486025.2017.1373201. ISSN 1748-6025. https://www.tandfonline.com/doi/full/10.1080/17486025.2017.1373201.
4. ↑ Ourabah, Kamel; Tribeche, Mouloud (2014). "Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics" (in en). Physical Review E 89 (6). doi:10.1103/PhysRevE.89.062130. ISSN 1539-3755. PMID 25019747. Bibcode: 2014PhRvE..89f2130O. https://link.aps.org/doi/10.1103/PhysRevE.89.062130.
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" (in en). The European Physical Journal Plus 136 (5): 518. doi:10.1140/epjp/s13360-021-01521-w. ISSN 2190-5444. Bibcode: 2021EPJP..136..518D. https://link.springer.com/10.1140/epjp/s13360-021-01521-w.
6. ↑ Macedo-Filho, A.; Moreira, D.A.; Silva, R.; da Silva, Luciano R. (2013). "Maximum entropy principle for Kaniadakis statistics and networks" (in en). Physics Letters A 377 (12): 842–846. doi:10.1016/j.physleta.2013.01.032. Bibcode: 2013PhLA..377..842M.

## External links

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

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