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.

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_{_{\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 $$\kappa \rightarrow 0$$.

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^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 $$\kappa \rightarrow 0$$.

Probability density function
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 $$0 \leq |\kappa| < 1$$ 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 $$\kappa \rightarrow 0$$.

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) $|undefined"with $$0 \leq F_\kappa \leq 1$$, in which the median is the case :"$x_{\textrm{median}} (F_\kappa) = \beta^{-1} \ln_\kappa (2) $|undefined"

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.