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
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 κ ( x ) = ( 1 − κ 2 ) β exp κ ( − β x ) {\displaystyle f_{_{\kappa }}(x)=(1-\kappa ^{2})\beta \exp _{\kappa }(-\beta x)}
valid for x ≥ 0 {\displaystyle x\geq 0} , where 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy and β > 0 {\displaystyle \beta >0} is known as rate parameter. The exponential distribution is recovered as κ → 0. {\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function The cumulative distribution function of κ-exponential distribution of Type I is given by
F κ ( x ) = 1 − ( 1 + κ 2 β 2 x 2 + κ 2 β x ) exp k ( − β x ) {\displaystyle F_{\kappa }(x)=1-{\Big (}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2}}}+\kappa ^{2}\beta x{\Big )}\exp _{k}({-\beta x)}}
for x ≥ 0 {\displaystyle x\geq 0} . The cumulative exponential distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
Properties
Moments, expectation value and variance The κ-exponential distribution of type I has moment of order m ∈ N {\displaystyle m\in \mathbb {N} } given by
E [ X m ] = 1 − κ 2 ∏ n = 0 m + 1 [ 1 − ( 2 n − m − 1 ) κ ] m ! β m {\displaystyle \operatorname {E} [X^{m}]={\frac {1-\kappa ^{2}}{\prod _{n=0}^{m+1}[1-(2n-m-1)\kappa ]}}{\frac {m!}{\beta ^{m}}}}
where f κ ( x ) {\displaystyle f_{\kappa }(x)} is finite if 0 < m + 1 < 1 / κ {\displaystyle 0<m+1<1/\kappa } . The expectation is defined as:
E [ X ] = 1 β 1 − κ 2 1 − 4 κ 2 {\displaystyle \operatorname {E} [X]={\frac {1}{\beta }}{\frac {1-\kappa ^{2}}{1-4\kappa ^{2}}}}
and the variance is:
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