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.^[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	[math]\displaystyle{ 0 \lt \kappa \lt 1 }[/math] shape (real) [math]\displaystyle{ \beta\gt 0 }[/math] rate (real)
Support	[math]\displaystyle{ x \in [0, \infty) }[/math]
PDF	[math]\displaystyle{ (1 - \kappa^2) \beta \exp_\kappa(-\beta x) }[/math]
CDF	[math]\displaystyle{ 1-\Big(\sqrt{1+\kappa^2\beta^2 x^2} + \kappa^2 \beta x \Big)\exp_k({-\beta x)} }[/math]
Mean	[math]\displaystyle{ \frac{1}{\beta} \frac{1 - \kappa^2}{1 - 4\kappa^2} }[/math]
Variance	[math]\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)} }[/math]
Skewness	[math]\displaystyle{ \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) } }[/math]
Kurtosis	[math]\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 }[/math]

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]

[math]\displaystyle{ f_{_{\kappa}}(x) = (1 - \kappa^2) \beta \exp_\kappa(-\beta x) }[/math]

valid for [math]\displaystyle{ x \ge 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy and [math]\displaystyle{ \beta \gt 0 }[/math] is known as rate parameter. The exponential distribution is recovered as [math]\displaystyle{ \kappa \rightarrow 0. }[/math]

Cumulative distribution function

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

[math]\displaystyle{ F_\kappa(x) = 1-\Big(\sqrt{1+\kappa^2\beta^2 x^2} + \kappa^2 \beta x \Big)\exp_k({-\beta x)} }[/math]

for [math]\displaystyle{ x \ge 0 }[/math]. The cumulative exponential distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Properties

Moments, expectation value and variance

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

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

where [math]\displaystyle{ f_\kappa(x) }[/math] is finite if [math]\displaystyle{ 0 \lt m + 1 \lt 1/\kappa }[/math].

The expectation is defined as:

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

and the variance is:

[math]\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)} }[/math]

Kurtosis

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

[math]\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] }[/math]

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

[math]\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 \lt 1/5 }[/math]

or

[math]\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 \lt 1/5 }[/math]

The kurtosis of the ordinary exponential distribution is recovered in the limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Skewness

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

[math]\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] }[/math]

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

[math]\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^3_\kappa (4 \kappa^2-1)^3 (144 \kappa^4-25 \kappa^2+1) } \quad \text{for} \quad 0 \leq \kappa \lt 1/4 }[/math]

The kurtosis of the ordinary exponential distribution is recovered in the limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Type II

Probability density function

κ-exponential distribution of type II

Probability density function
Cumulative distribution function
Parameters	[math]\displaystyle{ 0 \leq \kappa \lt 1 }[/math] shape (real) [math]\displaystyle{ \beta\gt 0 }[/math] rate (real)
Support	[math]\displaystyle{ x \in [0, \infty) }[/math]
PDF	[math]\displaystyle{ \frac{ \beta }{ \sqrt{1+ \kappa^2 \beta^2 x^2 } } \exp_\kappa(- \beta x) }[/math]
CDF	[math]\displaystyle{ 1-\exp_k({-\beta x)} }[/math]
Quantile	[math]\displaystyle{ \beta^{-1} \ln_\kappa \Bigg(\frac{1}{1 - F_\kappa} \Bigg) , 0 \leq F_\kappa \leq 1 }[/math]
Mean	[math]\displaystyle{ \frac{1}{\beta} \frac{1}{1 - \kappa^2} }[/math]
Median	[math]\displaystyle{ \beta^{-1} \ln_\kappa (2) }[/math]
Mode	[math]\displaystyle{ \frac{ 1 }{ \kappa \beta \sqrt{ 2 (1 - \kappa^2) } } }[/math]
Variance	[math]\displaystyle{ \sigma_\kappa^2 = \frac{1}{\beta^2} \frac{1+2 \kappa^4}{(1-4\kappa^2)(1-\kappa^2)^2} }[/math]
Skewness	[math]\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 } } }[/math]
Kurtosis	[math]\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) } }[/math]

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

[math]\displaystyle{ f_{_{\kappa}}(x) = \frac{\beta}{\sqrt{1+\kappa^2 \beta^2 x^2}} \exp_\kappa(-\beta x) }[/math]

valid for [math]\displaystyle{ x \ge 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy and [math]\displaystyle{ \beta \gt 0 }[/math] is known as rate parameter.

The exponential distribution is recovered as [math]\displaystyle{ \kappa \rightarrow 0. }[/math]

Cumulative distribution function

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

[math]\displaystyle{ F_\kappa(x) = 1-\exp_k({-\beta x)} }[/math]

for [math]\displaystyle{ x \ge 0 }[/math]. The cumulative exponential distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Properties

Moments, expectation value and variance

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

[math]\displaystyle{ \operatorname{E}[X^m] = \frac{\beta^{-m} m!}{\prod_{n=0}^{m} [1-(2n- m) \kappa ]} }[/math]

The expectation value and the variance are:

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

[math]\displaystyle{ \operatorname{Var}[X] = \sigma_\kappa^2 = \frac{1}{\beta^2} \frac{1+2 \kappa^4}{(1-4\kappa^2)(1-\kappa^2)^2} }[/math]

The mode is given by:

[math]\displaystyle{ x_{\textrm{mode}} = \frac{1}{\kappa \beta\sqrt{2(1-\kappa^2)}} }[/math]

Kurtosis

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

[math]\displaystyle{ \operatorname{Kurt}[X] = \operatorname{E}\left[\left(\frac{X - \frac{1}{\beta} \frac{1}{1 - \kappa^2} }{\sigma_\kappa} \right)^4 \right] }[/math]

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

[math]\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 \lt 1/4 }[/math]

or

[math]\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 \lt 1/4 }[/math]

Skewness

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

[math]\displaystyle{ \operatorname{Skew}[X] = \operatorname{E}\left[\frac{\left[ X - \frac{1}{\beta} \frac{1}{1 - \kappa^2}\right]^3}{\sigma_\kappa^3}\right] }[/math]

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

[math]\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 \lt 1/3 }[/math]

or

[math]\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 \lt 1/3 }[/math]

The skewness of the ordinary exponential distribution is recovered in the limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Quantiles

The quantiles are given by the following expression

[math]\displaystyle{ x_{\textrm{quantile}} (F_\kappa) = \beta^{-1} \ln_\kappa \Bigg(\frac{1}{1 - F_\kappa} \Bigg) }[/math]

with [math]\displaystyle{ 0 \leq F_\kappa \leq 1 }[/math], in which the median is the case :

[math]\displaystyle{ x_{\textrm{median}} (F_\kappa) = \beta^{-1} \ln_\kappa (2) }[/math]

Lorenz curve

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

[math]\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} }[/math]

The Gini coefficient is

[math]\displaystyle{ \operatorname{G}_\kappa = \frac{2 + \kappa^2}{4 - \kappa^2} }[/math]

Asymptotic behavior

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

[math]\displaystyle{ \lim_{x \to +\infty} f_\kappa (x) \sim \kappa^{-1} (2 \kappa \beta)^{-1/\kappa} x^{(-1 - \kappa)/\kappa} }[/math]

[math]\displaystyle{ \lim_{x \to 0^+} f_\kappa (x) = \beta }[/math]

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]

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

References

External links

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org

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