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

κ-exponential distribution of type I

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

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]

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

valid for ${\displaystyle x\geq 0}$, where ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index associated with the Kaniadakis entropy and ${\displaystyle \beta >0}$ is known as rate parameter. The exponential distribution is recovered as ${\displaystyle \kappa \rightarrow 0.}$

Cumulative distribution function

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

${\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2))}+\kappa ^{2}\beta x{\Big )}\exp _{k}({-\beta x)))$

for ${\displaystyle x\geq 0}$. The cumulative exponential distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

Properties

Moments, expectation value and variance

The κ-exponential distribution of type I has moment of order ${\displaystyle m\in \mathbb {N} }$ given by^[2]

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

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

The expectation is defined as:

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

and the variance is:

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

Kurtosis

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

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

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

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

or

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

The kurtosis of the ordinary exponential distribution is recovered in the limit ${\displaystyle \kappa \rightarrow 0}$.

Skewness

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

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

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

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

The kurtosis of the ordinary exponential distribution is recovered in the limit ${\displaystyle \kappa \rightarrow 0}$.

Type II

Probability density function

κ-exponential distribution of type II

Probability density function
Cumulative distribution function
Parameters	${\displaystyle 0\leq \kappa <1}$ shape (real) ${\displaystyle \beta >0}$ rate (real)
Support	${\displaystyle x\in [0,\infty )}$
PDF	${\displaystyle {\frac {\beta }{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2))))\exp _{\kappa }(-\beta x)}$
CDF	${\displaystyle 1-\exp _{k}({-\beta x)))$
Quantile	${\displaystyle \beta ^{-1}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa ))}{\Bigg )},0\leq F_{\kappa }\leq 1}$
Mean	${\displaystyle {\frac {1}{\beta )){\frac {1}{1-\kappa ^{2))))$
Median	${\displaystyle \beta ^{-1}\ln _{\kappa }(2)}$
Mode	${\displaystyle {\frac {1}{\kappa \beta {\sqrt {2(1-\kappa ^{2})))))}$
Variance	${\displaystyle \sigma _{\kappa }^{2}={\frac {1}{\beta ^{2))}{\frac {1+2\kappa ^{4)){(1-4\kappa ^{2})(1-\kappa ^{2})^{2))))$
Skewness	${\displaystyle {\frac {2(15\kappa ^{6}+6\kappa ^{4}+2\kappa ^{2}+1)}{(1-9\kappa ^{2})(2\kappa ^{4}+1))){\sqrt {\frac {1-4\kappa ^{2)){1+2\kappa ^{4))))}$
Excess kurtosis	${\displaystyle {\frac {3(72\kappa ^{10}-360\kappa ^{8}-44\kappa ^{6}-32\kappa ^{4}+7\kappa ^{2}-3)}{(4\kappa ^{2}-1)^{-1}(2\kappa ^{4}+1)^{2}(144\kappa ^{4}-25\kappa ^{2}+1)))}$
Method of Moments	${\displaystyle {\frac {\beta ^{-m}m!}{\prod _{n=0}^{m}[1-(2n-m)\kappa ]))}$

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 ${\displaystyle \alpha =1}$ is:^[2]

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

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

The exponential distribution is recovered as ${\displaystyle \kappa \rightarrow 0.}$

Cumulative distribution function

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

${\displaystyle F_{\kappa }(x)=1-\exp _{k}({-\beta x)))$

for ${\displaystyle x\geq 0}$. The cumulative exponential distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

Properties

Moments, expectation value and variance

The κ-exponential distribution of type II has moment of order ${\displaystyle m<1/\kappa }$ given by^[2]

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

The expectation value and the variance are:

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

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

The mode is given by:

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

Kurtosis

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

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

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

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

or

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

Skewness

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

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

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

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

or

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

The skewness of the ordinary exponential distribution is recovered in the limit ${\displaystyle \kappa \rightarrow 0}$.

Quantiles

The quantiles are given by the following expression

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

with ${\displaystyle 0\leq F_{\kappa }\leq 1}$, in which the median is the case :

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

Lorenz curve

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

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

The Gini coefficient is

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

Asymptotic behavior

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

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

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

Applications

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

• In geomechanics, for analyzing the properties of rock masses;^[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]