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.{{Cite journal |last=Kaniadakis |first=G. |date=2001 |title=Non-linear kinetics underlying generalized statistics |url=https://linkinghub.elsevier.com/retrieve/pii/S0378437101001844 |journal=Physica A: Statistical Mechanics and Its Applications |language=en |volume=296 |issue=3–4 |pages=405–425 |doi=10.1016/S0378-4371(01)00184-4|arxiv=cond-mat/0103467 |bibcode=2001PhyA..296..405K |s2cid=44275064 }} 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=

{{Probability distribution

| name = κ-exponential distribution of type I

| type = density

| parameters = $0 < \kappa < 1$ shape (real)
$\beta> 0$ rate (real)

| support = $x \in [0, \infty)$

| pdf = $(1 - \kappa^2) \beta \exp_\kappa(-\beta x)$

| cdf = $1-\Big(\sqrt{1+\kappa^2\beta^2 x^2} + \kappa^2 \beta x \Big)\exp_k({-\beta x)}$

| mean = $\frac{1}{\beta} \frac{1 - \kappa^2}{1 - 4\kappa^2}$

| variance = $\sigma_\kappa^2 =\frac{1}{\beta^2} \frac{2(1-4\kappa^2)^2 - (1 - \kappa^2)^2(1-9\kappa^2)}{(1-4\kappa^2)^2(1-9\kappa^2)}$

|pdf_image=File:Kaniadakis Exponential Distribution Type I pdf.png|cdf_image=File:Kaniadakis Exponential Distribution Type I cdf.png|moments= $\frac{1 - \kappa^2}{\prod_{n=0}^{m+1} [1-(2n-m-1) \kappa ]} \frac{m!}{\beta^m}$ |kurtosis= $\frac{ 9(1200\kappa^{14} - 6123\kappa^{12} + 562\kappa^{10} +1539 \kappa^8 - 544 \kappa^6 + 143 \kappa^4 -18\kappa^2 + 1 )}{ \beta^4 \sigma_\kappa^4 (1-\kappa^2)^{-1}(1 - 4\kappa^2)^4 (3600\kappa^8 -4369\kappa^6 + 819\kappa^4 - 51\kappa^2 + 1) } - 3$ |skewness= $\frac{ 2 (1-\kappa^2) (144 \kappa^8+23 \kappa^6+27 \kappa^4-6 \kappa^2+1) }{ \beta^3 \sigma^3_\kappa (4 \kappa^2-1)^3 (144 \kappa^4-25 \kappa^2+1) }$ }}

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:{{Cite journal |last=Kaniadakis |first=G. |date=2021-01-01 |title=New power-law tailed distributions emerging in κ-statistics (a) |url=https://iopscience.iop.org/article/10.1209/0295-5075/133/10002 |journal=Europhysics Letters |volume=133 |issue=1 |pages=10002 |doi=10.1209/0295-5075/133/10002 |issn=0295-5075|arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 }}

: $f_{_{\kappa}}(x) = (1 - \kappa^2) \beta \exp_\kappa(-\beta x)$

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 $\kappa \rightarrow 0.$

=Cumulative distribution function=

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

: $F_\kappa(x) = 1-\Big(\sqrt{1+\kappa^2\beta^2 x^2} + \kappa^2 \beta x \Big)\exp_k({-\beta x)}$

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

= Properties =

== Moments, expectation value and variance ==

The κ-exponential distribution of type I has moment of order $m \in \mathbb{N}$ given by

: $\operatorname{E}[X^m] = \frac{1 - \kappa^2}{\prod_{n=0}^{m+1} [1-(2n-m-1) \kappa ]} \frac{m!}{\beta^m}$

where $f_\kappa(x)$ is finite if $0 < m + 1 < 1/\kappa$ .

The expectation is defined as:

: $\operatorname{E}[X] = \frac{1}{\beta} \frac{1 - \kappa^2}{1 - 4\kappa^2}$

and the variance is:

: $\operatorname{Var}[X] = \sigma_\kappa^2 = \frac{1}{\beta^2} \frac{2(1-4\kappa^2)^2 - (1 - \kappa^2)^2(1-9\kappa^2)}{(1-4\kappa^2)^2(1-9\kappa^2)}$

== Kurtosis ==

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

: $\operatorname{Kurt}[X] = \operatorname{E}\left[\frac{\left[ X - \frac{1}{\beta} \frac{1 - \kappa^2}{1 - 4\kappa^2}\right]^4}{\sigma_\kappa^4}\right]$

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

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

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

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

\kappa \rightarrow 0

== Skewness ==

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

: $\operatorname{Skew}[X] = \operatorname{E}\left[\frac{\left[ X - \frac{1}{\beta} \frac{1 - \kappa^2}{1 - 4\kappa^2}\right]^3}{\sigma_\kappa^3}\right]$

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

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

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

\kappa \rightarrow 0

Type II

=Probability density function=

{{Probability distribution

| name = κ-exponential distribution of type II

| type = density

| parameters = $0 \leq \kappa < 1$ shape (real)
$\beta> 0$ rate (real)

| support = $x \in [0, \infty)$

| pdf = $\frac{ \beta }{ \sqrt{1+ \kappa^2 \beta^2 x^2 } }
\exp_\kappa(- \beta x)$

| cdf = $1-\exp_k({-\beta x)}$

| mean = $\frac{1}{\beta} \frac{1}{1 - \kappa^2}$

| variance = $\sigma_\kappa^2 = \frac{1}{\beta^2} \frac{1+2 \kappa^4}{(1-4\kappa^2)(1-\kappa^2)^2}$

|pdf_image=File:Kaniadakis Exponential Distribution Type II pdf.png|cdf_image=File:Kaniadakis Exponential Distribution Type II cdf.png|mode= $\frac{ 1 }{ \kappa \beta \sqrt{ 2 (1 - \kappa^2) } }$ |quantile= $\beta^{-1} \ln_\kappa \Bigg(\frac{1}{1 - F_\kappa} \Bigg) , 0 \leq F_\kappa \leq 1$ |median= $\beta^{-1} \ln_\kappa (2)$ |moments= $\frac{\beta^{-m} m!}{\prod_{n=0}^{m} [1-(2n- m) \kappa ]}$ |kurtosis= $\frac{3 (72 \kappa^{10} - 360 \kappa^8 - 44 \kappa^6-32 \kappa^4+7 \kappa^2-3) }{ (4\kappa^2-1)^{-1} (2 \kappa^4+1)^2 (144 \kappa^4-25 \kappa^2+1) }$ |skewness= $\frac{ 2 (15 \kappa^6+6 \kappa^4+2 \kappa^2+1) }{ (1 - 9\kappa^2)(2 \kappa^4 + 1) } \sqrt{ \frac{1 - 4\kappa^2 }{ 1 + 2\kappa^4 } }$ }}

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:{{Cite journal |last=Kaniadakis |first=G. |date=2021-01-01 |title=New power-law tailed distributions emerging in κ-statistics (a) |url=https://iopscience.iop.org/article/10.1209/0295-5075/133/10002 |journal=Europhysics Letters |volume=133 |issue=1 |pages=10002 |doi=10.1209/0295-5075/133/10002 |issn=0295-5075|arxiv=2203.01743 |bibcode=2021EL....13310002K |s2cid=234144356 }}

: $f_{_{\kappa}}(x) =
\frac{\beta}{\sqrt{1+\kappa^2 \beta^2 x^2}}
\exp_\kappa(-\beta x)$

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 $\kappa \rightarrow 0.$

=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 $\kappa \rightarrow 0$ .

= Properties =

== Moments, expectation value and variance ==

The κ-exponential distribution of type II has moment of order $m < 1/\kappa$ given by

: $\operatorname{E}[X^m] = \frac{\beta^{-m} m!}{\prod_{n=0}^{m} [1-(2n- m) \kappa ]}$

The expectation value and the variance are:

: $\operatorname{E}[X] = \frac{1}{\beta} \frac{1}{1 - \kappa^2}$

: $\operatorname{Var}[X] = \sigma_\kappa^2 = \frac{1}{\beta^2} \frac{1+2 \kappa^4}{(1-4\kappa^2)(1-\kappa^2)^2}$

The mode is given by:

: $x_{\textrm{mode}} = \frac{1}{\kappa \beta\sqrt{2(1-\kappa^2)}}$

== Kurtosis ==

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

: $\operatorname{Kurt}[X] = \operatorname{E}\left[\left(\frac{X - \frac{1}{\beta} \frac{1}{1 - \kappa^2} }{\sigma_\kappa} \right)^4 \right]$

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

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

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

== Skewness ==

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

: $\operatorname{Skew}[X] = \operatorname{E}\left[\frac{\left[ X - \frac{1}{\beta} \frac{1}{1 - \kappa^2}\right]^3}{\sigma_\kappa^3}\right]$

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

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

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

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

\kappa \rightarrow 0

== Quantiles ==

The quantiles are given by the following expression

$x_{\textrm{quantile}} (F_\kappa) = \beta^{-1} \ln_\kappa \Bigg(\frac{1}{1 - F_\kappa} \Bigg)$

with

0 \leq F_\kappa \leq 1

, in which the median is the case :

$x_{\textrm{median}} (F_\kappa) = \beta^{-1} \ln_\kappa (2)$

== Lorenz curve ==

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

: $\mathcal{L}_\kappa(F_\kappa) = 1 + \frac{1 - \kappa}{2 \kappa}(1 - F_\kappa)^{1 + \kappa} - \frac{1 + \kappa}{2 \kappa}(1 - F_\kappa)^{1 - \kappa}$

The Gini coefficient is

$\operatorname{G}_\kappa = \frac{2 + \kappa^2}{4 - \kappa^2}$

== Asymptotic behavior ==

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

: $\lim_{x \to +\infty} f_\kappa (x) \sim \kappa^{-1} (2 \kappa \beta)^{-1/\kappa} x^{(-1 - \kappa)/\kappa}$

: $\lim_{x \to 0^+} f_\kappa (x) = \beta$

Applications

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

In geomechanics, for analyzing the properties of rock masses;{{Cite journal |last1=Oreste |first1=Pierpaolo |last2=Spagnoli |first2=Giovanni |date=2018-04-03 |title=Statistical analysis of some main geomechanical formulations evaluated with the Kaniadakis exponential law |url=https://www.tandfonline.com/doi/full/10.1080/17486025.2017.1373201 |journal=Geomechanics and Geoengineering |language=en |volume=13 |issue=2 |pages=139–145 |doi=10.1080/17486025.2017.1373201 |s2cid=133860553 |issn=1748-6025|url-access=subscription }}
In quantum theory, in physical analysis using Planck's radiation law;{{Cite journal |last1=Ourabah |first1=Kamel |last2=Tribeche |first2=Mouloud |date=2014 |title=Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics |url=https://link.aps.org/doi/10.1103/PhysRevE.89.062130 |journal=Physical Review E |language=en |volume=89 |issue=6 |pages=062130 |doi=10.1103/PhysRevE.89.062130 |pmid=25019747 |bibcode=2014PhRvE..89f2130O |issn=1539-3755|url-access=subscription }}
In inverse problems, the κ-exponential distribution has been used to formulate a robust approach;{{Cite journal |last1=da Silva |first1=Sérgio Luiz E. F. |last2=dos Santos Lima |first2=Gustavo Z. |last3=Volpe |first3=Ernani V. |last4=de Araújo |first4=João M. |last5=Corso |first5=Gilberto |date=2021 |title=Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics |url=https://link.springer.com/10.1140/epjp/s13360-021-01521-w |journal=The European Physical Journal Plus |language=en |volume=136 |issue=5 |pages=518 |doi=10.1140/epjp/s13360-021-01521-w |bibcode=2021EPJP..136..518D |s2cid=236575441 |issn=2190-5444|url-access=subscription }}
In Network theory.{{Cite journal |last1=Macedo-Filho |first1=A. |last2=Moreira |first2=D.A. |last3=Silva |first3=R. |last4=da Silva |first4=Luciano R. |date=2013 |title=Maximum entropy principle for Kaniadakis statistics and networks |journal=Physics Letters A |language=en |volume=377 |issue=12 |pages=842–846 |doi=10.1016/j.physleta.2013.01.032|bibcode=2013PhLA..377..842M |doi-access=free }}

References

External links

[https://scholar.google.it/citations?user=pFlYesUAAAAJ&hl=en Giorgio Kaniadakis Google Scholar page]
[https://arxiv.org/search/?query=kaniadakis+statistics&searchtype=all&abstracts=show&order=-announced_date_first&size=200 Kaniadakis Statistics on arXiv.org]

Category:Probability distributions

Category:Mathematical and quantitative methods (economics)

Category:Continuous distributions

Category:Exponentials

Category:Exponential family distributions