Kaniadakis exponential distribution explained

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

Probability density function

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]

f _{_{\kappa}} (x) = ( 1 - \kappa ^{2)} \beta \exp _{\kappa(-\beta} x)

valid for

x \ge 0 , where 0 \leq |\kappa| < 1 is the entropic index associated with the Kaniadakis entropy and \beta > 0 is known as rate parameter. The exponential distribution is recovered as \kappa → 0 .

Cumulative distribution function

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

F _{\kappa(x)} = 1- ( \sqrt{ 1+ \kappa ^{2\beta} ^{2} x ^{2}} + \kappa ^{2} \beta x ) \exp _{k({-\beta} x)}

for

x \ge 0 . The cumulative exponential distribution is recovered in the classical limit \kappa → 0 .

Properties

Moments, expectation value and variance

The κ-exponential distribution of type I has moment of order

m \in N given by\operatorname{E}[X ^{m]} = (1- \kappa ^{2})/(\prod ^{m +1} _{n =0} [ 1- ( 2 n - m -1 ) \kappa ]) (m!)/(\beta ^{m})

where

f _{\kappa(x)} is finite if 0 < m + 1 < 1 /\kappa .

The expectation is defined as:

\operatorname{E}[X] = (1)/(\beta) (1- \kappa ^{2})/(1-4 \kappa ^{2}) and the variance is:\operatorname{Var}[X] = \sigma ^{2} _{\kappa} = (1)/(\beta ^{2}) (2 ( 1-4 \kappa ^{2)} ^{2} - ( 1- \kappa ^{2)} ^{2(1-9\kappa} ^{2)})/(( 1-4 \kappa ^{2)} ^{2(1-9\kappa} ^{2)})

Kurtosis

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

\operatorname{Kurt}[X] = \operatorname{E}\left[ (\left[ X - (1)/(\beta) (1- \kappa ^{2})/(1-4 \kappa ^{2}) \right] ^{4})/(\sigma ^{4} _{\kappa}) \right] Thus, the kurtosis of the κ-exponential distribution of type I distribution is given by:

\operatorname{Kurt}[X] = (9 ( 1- \kappa ^{2)(1200\kappa} ^{14} -6123 \kappa ^{12} +562 \kappa ^{10} +1539 \kappa ^{8} -544 \kappa ^{6} +143 \kappa ^{4} -18 \kappa ^{2} +1 ))/(\beta ^{4} \sigma ^{4} _{\kappa} ( 1 - 4 \kappa ^{2)} ^{4} ( 3600 \kappa ^{8} -4369 \kappa ^{6} + 819 \kappa ^{4} - 51 \kappa ^{2} + 1 )) for 0 \leq \kappa < 1 / 5

or

\operatorname{Kurt}[X] = (9 ( 9 \kappa ^{2-1)} ^{2(\kappa} ^{2-1)(1200\kappa} ^{14} -6123 \kappa ^{12} +562 \kappa ^{10} +1539 \kappa ^{8} -544 \kappa ^{6} +143 \kappa ^{4} -18 \kappa ^{2} +1 ))/(\beta ^{2} ( 1-4 \kappa ^{2)} ^{2(9\kappa} ^{6} +13 \kappa ^{4} -5 \kappa ^{2} +1 )( 3600 \kappa ^{8} -4369 \kappa ^{6} +819 \kappa ^{4} -51 \kappa ^{2} +1 )) for 0 \leq \kappa < 1 / 5

The kurtosis of the ordinary exponential distribution is recovered in the limit \kappa → 0 .

Skewness

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

\operatorname{Skew}[X] = \operatorname{E}\left[ (\left[ X - (1)/(\beta) (1- \kappa ^{2})/(1-4 \kappa ^{2}) \right] ^{3})/(\sigma ^{3} _{\kappa}) \right] Thus, the skewness of the κ-exponential distribution of type I distribution is given by:

\operatorname{Shew}[X] = (2 ( 1- \kappa ^{2)} ( 144 \kappa ^{8+23} \kappa ^{6+27} \kappa ^{4-6} \kappa ^{2+1)})/(\beta ^{3} \sigma ^{3} _{\kappa} ( 4 \kappa ^{2-1)} ^{3} ( 144 \kappa ^{4-25} \kappa ^{2+1)}) for 0 \leq \kappa < 1 / 4

The kurtosis of the ordinary exponential distribution is recovered in the limit \kappa → 0 .

Type II

Probability density function

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

\alpha = 1 is:^[3] f _{_{\kappa}} (x) = (\beta)/(\sqrt{ 1+ \kappa ^{2} \beta ^{2} x ^{2}) }\exp_\kappa(-\beta x)

valid for

x \ge 0 , where 0 \leq |\kappa| < 1 is the entropic index associated with the Kaniadakis entropy and \beta > 0 is known as rate parameter.

The exponential distribution is recovered as

\kappa → 0 .

Cumulative distribution function

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

F _{\kappa(x)} = 1- \exp _{k({-\beta} x)}

for

x \ge 0 . The cumulative exponential distribution is recovered in the classical limit \kappa → 0 .

Properties

Moments, expectation value and variance

The κ-exponential distribution of type II has moment of order

m < 1 /\kappa given by\operatorname{E}[X ^{m]} = (\beta ^{-m} m!)/(\prod ^{m} _{n =0} [ 1- ( 2 n - m) \kappa ])

The expectation value and the variance are:

\operatorname{E}[X] = (1)/(\beta) (1)/(1- \kappa ^{2}) \operatorname{Var}[X] = \sigma ^{2} _{\kappa} = (1)/(\beta ^{2}) (1+2 \kappa ^{4})/(( 1-4 \kappa ^{2)(1-\kappa} ^{2)} ^{2})

The mode is given by:

x _{rm{mode} } = \frac

Kurtosis

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

\operatorname{Kurt}[X] = \operatorname{E}\left[\left( (X - (1)/(\beta) (1)/(1- \kappa ^{2}))/(\sigma _{\kappa}) \right) ^{4} \right]

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

\operatorname{Kurt}[X] = (3 ( 72 \kappa ^{10} -360 \kappa ^{8} -44 \kappa ^{6-32} \kappa ^{4+7} \kappa ^{2-3)})/(\beta ^{4} \sigma ^{4} _{\kappa} (\kappa ^{2} - 1 ) ^{4} ( 576 \kappa ^{6} - 244 \kappa ^{4} + 29 \kappa ^{2} - 1 )) for 0 \leq \kappa < 1 / 4

or

\operatorname{Kurt}[X] = (3 ( 72 \kappa ^{10} -360 \kappa ^{8} -44 \kappa ^{6-32} \kappa ^{4+7} \kappa ^{2-3)})/(( 4 \kappa ^{2-1)} ^{-1} ( 2 \kappa ^{4+1)} ^{2} ( 144 \kappa ^{4-25} \kappa ^{2+1)}) for 0 \leq \kappa < 1 / 4

Skewness

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

\operatorname{Skew}[X] = \operatorname{E}\left[ (\left[ X - (1)/(\beta) (1)/(1- \kappa ^{2}) \right] ^{3})/(\sigma ^{3} _{\kappa}) \right] Thus, the skewness of the κ-exponential distribution of type II distribution is given by:

\operatorname{Skew}[X] =- (2 ( 15 \kappa ^{6+6} \kappa ^{4+2} \kappa ^{2+1)})/(\beta ^{3} \sigma ^{3} _{\kappa} (\kappa ^{2} - 1 ) ^{3} ( 36 \kappa ^{4} - 13 \kappa ^{2} + 1 )) for 0 \leq \kappa < 1 / 3

or

\operatorname{Skew}[X] = (2 ( 15 \kappa ^{6+6} \kappa ^{4+2} \kappa ^{2+1)})/(( 1-9 \kappa ^{2)(2} \kappa ^{4} +1 )) \sqrt{ (1-4 \kappa ^{2})/(1+2 \kappa ^{4}) } for 0 \leq \kappa < 1 / 3

The skewness of the ordinary exponential distribution is recovered in the limit \kappa → 0 .

Quantiles

The quantiles are given by the following expression

x _{rm{quantile} } (F_\kappa) = \beta^ \ln_\kappa \Bigg(\frac \Bigg)

with 0 \leq F _{\kappa} \leq 1 , in which the median is the case :

x _{rm{median} } (F_\kappa) = \beta^ \ln_\kappa (2)

Lorenz curve

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

l{L} _{\kappa(F} _{\kappa)} = 1 + (1- \kappa)/(2 \kappa) ( 1 - F ^{1+ \kappa} _{\kappa)} - (1+ \kappa)/(2 \kappa) ( 1 - F ^{1- \kappa} _{\kappa)} The Gini coefficient is

\operatorname{G} _{\kappa} = (2+ \kappa ^{2})/(4- \kappa ^{2})

Asymptotic behavior

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

\lim _{x} f _{\kappa} (x) \sim \kappa ^{-1} ( 2 \kappa \beta) ^{-1/\kappa} x ^{(-1} \lim _{x \to 0 ^{+}} f _{\kappa} (x) = \beta

Applications

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

• In geomechanics, for analyzing the properties of rock masses;^[4]
• In quantum theory, in physical analysis using Planck's radiation law;^[5]
• In inverse problems, the κ-exponential distribution has been used to formulate a robust approach;^[6]
• In Network theory.^[7]

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

External links

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

Notes and References

1. Kaniadakis . G. . 2001 . Non-linear kinetics underlying generalized statistics . Physica A: Statistical Mechanics and Its Applications . en . 296 . 3–4 . 405–425 . 10.1016/S0378-4371(01)00184-4. cond-mat/0103467 . 2001PhyA..296..405K . 44275064 .
2. Kaniadakis . G. . 2021-01-01 . New power-law tailed distributions emerging in κ-statistics (a) . Europhysics Letters . 133 . 1 . 10002 . 10.1209/0295-5075/133/10002 . 0295-5075. 2203.01743 . 2021EL....13310002K . 234144356 .
3. Kaniadakis . G. . 2021-01-01 . New power-law tailed distributions emerging in κ-statistics (a) . Europhysics Letters . 133 . 1 . 10002 . 10.1209/0295-5075/133/10002 . 0295-5075. 2203.01743 . 2021EL....13310002K . 234144356 .
4. Oreste . Pierpaolo . Spagnoli . Giovanni . 2018-04-03 . Statistical analysis of some main geomechanical formulations evaluated with the Kaniadakis exponential law . Geomechanics and Geoengineering . en . 13 . 2 . 139–145 . 10.1080/17486025.2017.1373201 . 133860553 . 1748-6025.
5. Ourabah . Kamel . Tribeche . Mouloud . 2014 . Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics . Physical Review E . en . 89 . 6 . 062130 . 10.1103/PhysRevE.89.062130 . 25019747 . 2014PhRvE..89f2130O . 1539-3755.
6. 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 . en . 136 . 5 . 518 . 10.1140/epjp/s13360-021-01521-w . 2021EPJP..136..518D . 236575441 . 2190-5444.
7. Macedo-Filho . A. . Moreira . D.A. . Silva . R. . da Silva . Luciano R. . 2013 . Maximum entropy principle for Kaniadakis statistics and networks . Physics Letters A . en . 377 . 12 . 842–846 . 10.1016/j.physleta.2013.01.032. 2013PhLA..377..842M . free .