Kaniadakis exponential 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.
Type I edit
Probability density function edit
Probability density function | |||
Cumulative distribution function | |||
Parameters |
shape (real) rate (real) | ||
---|---|---|---|
Support | |||
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 edit
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 edit
Moments, expectation value and variance edit
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 edit
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 edit
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 edit
Probability density function edit
Probability density function | |||
Cumulative distribution function | |||
Parameters |
shape (real) rate (real) | ||
---|---|---|---|
Support | |||
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 edit
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 edit
Moments, expectation value and variance edit
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 edit
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 edit
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 edit
The quantiles are given by the following expression
with , in which the median is the case :
Lorenz curve edit
The Lorenz curve associated with the κ-exponential distribution of type II is given by:[2]
The Gini coefficient is
Asymptotic behavior edit
The κ-exponential distribution of type II behaves asymptotically as follows:[2]
Applications edit
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 edit
References edit
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.