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. 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	$x\in [0,\infty )$
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	$\sigma _{\kappa }^{2}={\frac {1}{\beta ^{2}}}{\frac {2(1-4\kappa ^{2})^{2}-(1-\kappa ^{2})^{2}(1-9\kappa ^{2})}{(1-4\kappa ^{2})^{2}(1-9\kappa ^{2})}}$
Skewness	${\frac {2(1-\kappa ^{2})(144\kappa ^{8}+23\kappa ^{6}+27\kappa ^{4}-6\kappa ^{2}+1)}{\beta ^{3}\sigma _{\kappa }^{3}(4\kappa ^{2}-1)^{3}(144\kappa ^{4}-25\kappa ^{2}+1)}}$
Excess kurtosis	${\frac {9(1200\kappa ^{14}-6123\kappa ^{12}+562\kappa ^{10}+1539\kappa ^{8}-544\kappa ^{6}+143\kappa ^{4}-18\kappa ^{2}+1)}{\beta ^{4}\sigma _{\kappa }^{4}(1-\kappa ^{2})^{-1}(1-4\kappa ^{2})^{4}(3600\kappa ^{8}-4369\kappa ^{6}+819\kappa ^{4}-51\kappa ^{2}+1)}}-3$
Method of moments	${\frac {1-\kappa ^{2}}{\prod _{n=0}^{m+1}[1-(2n-m-1)\kappa ]}}{\frac {m!}{\beta ^{m}}}$

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:

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

valid for $x\geq 0$, 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 $x\geq 0$. 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:

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

or

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

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:

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

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

Type II

Probability density function

κ-exponential distribution of type II

Probability density function
Cumulative distribution function
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	$1-\exp _{k}({-\beta x)}$
Quantile	$\beta ^{-1}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )},0\leq F_{\kappa }\leq 1$
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	${\frac {2(15\kappa ^{6}+6\kappa ^{4}+2\kappa ^{2}+1)}{(1-9\kappa ^{2})(2\kappa ^{4}+1)}}{\sqrt {\frac {1-4\kappa ^{2}}{1+2\kappa ^{4}}}}$
Excess kurtosis	${\frac {3(72\kappa ^{10}-360\kappa ^{8}-44\kappa ^{6}-32\kappa ^{4}+7\kappa ^{2}-3)}{(4\kappa ^{2}-1)^{-1}(2\kappa ^{4}+1)^{2}(144\kappa ^{4}-25\kappa ^{2}+1)}}$
Method of moments	${\frac {\beta ^{-m}m!}{\prod _{n=0}^{m}[1-(2n-m)\kappa ]}}$

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 $\alpha =1$ is:

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

valid for $x\geq 0$, 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 $x\geq 0$. 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:

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

or

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

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:

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

or

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

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;
• 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