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. 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

κ-exponential distribution of type I

Probability density function
Cumulative distribution function
Parameters	$0<\kappa <1$ shape (real) $\beta >0$ rate (real)
Support	$x\in [0,\mathrm{\infty })$
PDF	$(1-{\kappa }^2)\beta {\exp }_{\kappa }(-\beta x)$
CDF	$1-(\sqrt{1+{\kappa }^2{\beta }^2{x}^2}+{\kappa }^2\beta x){\exp }_k(-\beta x)$
Mean	$\frac{1}{\beta }\frac{1-{\kappa }^2}{1-4{\kappa }^2}$
Variance	${\sigma }_{\kappa }^2=\frac{1}{{\beta }^2}\frac{2(1-4{\kappa }^2{)}^2-(1-{\kappa }^2{)}^2(1-9{\kappa }^2)}{(1-4{\kappa }^2{)}^2(1-9{\kappa }^2)}$
Skewness	$\frac{2(1-{\kappa }^2)(144{\kappa }^8+23{\kappa }^6+27{\kappa }^4-6{\kappa }^2+1)}{{\beta }^3{\sigma }_{\kappa }^3(4{\kappa }^2-1{)}^3(144{\kappa }^4-25{\kappa }^2+1)}$
Kurtosis	$\frac{9(1200{\kappa }^{14}-6123{\kappa }^{12}+562{\kappa }^{10}+1539{\kappa }^8-544{\kappa }^6+143{\kappa }^4-18{\kappa }^2+1)}{{\beta }^4{\sigma }_{\kappa }^4(1-{\kappa }^2{)}^{-1}(1-4{\kappa }^2{)}^4(3600{\kappa }^8-4369{\kappa }^6+819{\kappa }^4-51{\kappa }^2+1)}-3$

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:

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

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

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 $\kappa \to 0$.

Properties

Moments, expectation value and variance

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

$\mathrm{E}[X^m]=\frac{1-{\kappa }^2}{\prod _{n=0}^{m+1}[1-(2n-m-1)\kappa ]}\frac{m!}{{\beta }^m}$

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

The expectation is defined as:

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

and the variance is:

$\mathrm{Var}[X]={\sigma }_{\kappa }^2=\frac{1}{{\beta }^2}\frac{2(1-4{\kappa }^2{)}^2-(1-{\kappa }^2{)}^2(1-9{\kappa }^2)}{(1-4{\kappa }^2{)}^2(1-9{\kappa }^2)}$

Kurtosis

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

$\mathrm{Kurt}[X]=\mathrm{E}\left[\frac{{[X-\frac{1}{\beta }\frac{1-{\kappa }^2}{1-4{\kappa }^2}]}^4}{{\sigma }_{\kappa }^4}\right]$

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

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

or

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

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

$\kappa \to 0$

.

Skewness

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

$\mathrm{Skew}[X]=\mathrm{E}\left[\frac{{[X-\frac{1}{\beta }\frac{1-{\kappa }^2}{1-4{\kappa }^2}]}^3}{{\sigma }_{\kappa }^3}\right]$

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

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

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

$\kappa \to 0$

.

Type II

Probability density function

κ-exponential distribution of type II

Probability density function
Cumulative distribution function
Parameters	$0\le \kappa <1$ shape (real) $\beta >0$ rate (real)
Support	$x\in [0,\mathrm{\infty })$
PDF	$\frac{\beta }{\sqrt{1+{\kappa }^2{\beta }^2{x}^2}}{\exp }_{\kappa }(-\beta x)$
CDF	$1-{\exp }_k(-\beta x)$
Quantile	${\beta }^{-1}{\ln }_{\kappa }\left(\frac{1}{1-F_{\kappa }}\right),0\le F_{\kappa }\le 1$
Mean	$\frac{1}{\beta }\frac{1}{1-{\kappa }^2}$
Median	${\beta }^{-1}{\ln }_{\kappa }(2)$
Mode	$\frac{1}{\kappa \beta \sqrt{2(1-{\kappa }^2)}}$
Variance	${\sigma }_{\kappa }^2=\frac{1}{{\beta }^2}\frac{1+2{\kappa }^4}{(1-4{\kappa }^2)(1-{\kappa }^2{)}^2}$
Skewness	$\frac{2(15{\kappa }^6+6{\kappa }^4+2{\kappa }^2+1)}{(1-9{\kappa }^2)(2{\kappa }^4+1)}\sqrt{\frac{1-4{\kappa }^2}{1+2{\kappa }^4}}$
Kurtosis	$\frac{3(72{\kappa }^{10}-360{\kappa }^8-44{\kappa }^6-32{\kappa }^4+7{\kappa }^2-3)}{(4{\kappa }^2-1{)}^{-1}(2{\kappa }^4+1{)}^2(144{\kappa }^4-25{\kappa }^2+1)}$

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:

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

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

The exponential distribution is recovered as $\kappa \to 0.$

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 $\kappa \to 0$.

Properties

Moments, expectation value and variance

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

$\mathrm{E}[X^m]=\frac{{\beta }^{-m}m!}{\prod _{n=0}^m[1-(2n-m)\kappa ]}$

The expectation value and the variance are:

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

$\mathrm{Var}[X]={\sigma }_{\kappa }^2=\frac{1}{{\beta }^2}\frac{1+2{\kappa }^4}{(1-4{\kappa }^2)(1-{\kappa }^2{)}^2}$

The mode is given by:

$x_{\text{<mrow data-mjx-texclass="ORD">mode</mrow>}}=\frac{1}{\kappa \beta \sqrt{2(1-{\kappa }^2)}}$

Kurtosis

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

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

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

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

or

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

Skewness

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

$\mathrm{Skew}[X]=\mathrm{E}\left[\frac{{[X-\frac{1}{\beta }\frac{1}{1-{\kappa }^2}]}^3}{{\sigma }_{\kappa }^3}\right]$

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

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

or

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

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

$\kappa \to 0$

.

Quantiles

The quantiles are given by the following expression

$x_{\text{<mrow data-mjx-texclass="ORD">quantile</mrow>}}(F_{\kappa })={\beta }^{-1}{\ln }_{\kappa }\left(\frac{1}{1-F_{\kappa }}\right)$

with

$0\le F_{\kappa }\le 1$

, in which the median is the case :

$x_{\text{<mrow data-mjx-texclass="ORD">median</mrow>}}(F_{\kappa })={\beta }^{-1}{\ln }_{\kappa }(2)$

Lorenz curve

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

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

The Gini coefficient is

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

Asymptotic behavior

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

$\underset{x\to +\mathrm{\infty }}{\lim }{f}_{\kappa }(x)\sim {\kappa }^{-1}(2\kappa \beta {)}^{-1/\kappa }{x}^{(-1-\kappa )/\kappa }$

$\underset{x\to 0^{+}}{\lim }{f}_{\kappa }(x)=\beta$

Applications

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

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

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