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Kaniadakis exponential distribution

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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 < κ < 1 {\displaystyle 0<\kappa <1} shape (real)
β > 0 {\displaystyle \beta >0} rate (real)
Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )}
PDF ( 1 − κ 2 ) β exp κ ⁡ ( − β x ) {\displaystyle (1-\kappa ^{2})\beta \exp _{\kappa }(-\beta x)}
CDF 1 − ( 1 + κ 2 β 2 x 2 + κ 2 β x ) exp k ⁡ ( − β x ) {\displaystyle 1-{\Big (}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2}}}+\kappa ^{2}\beta x{\Big )}\exp _{k}({-\beta x)}}
Mean 1 β 1 − κ 2 1 − 4 κ 2 {\displaystyle {\frac {1}{\beta }}{\frac {1-\kappa ^{2}}{1-4\kappa ^{2}}}}
Variance σ κ 2 = 1 β 2 2 ( 1 − 4 κ 2 ) 2 − ( 1 − κ 2 ) 2 ( 1 − 9 κ 2 ) ( 1 − 4 κ 2 ) 2 ( 1 − 9 κ 2 ) {\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 2 ( 1 − κ 2 ) ( 144 κ 8 + 23 κ 6 + 27 κ 4 − 6 κ 2 + 1 ) β 3 σ κ 3 ( 4 κ 2 − 1 ) 3 ( 144 κ 4 − 25 κ 2 + 1 ) {\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 9 ( 1200 κ 14 − 6123 κ 12 + 562 κ 10 + 1539 κ 8 − 544 κ 6 + 143 κ 4 − 18 κ 2 + 1 ) β 4 σ κ 4 ( 1 − κ 2 ) − 1 ( 1 − 4 κ 2 ) 4 ( 3600 κ 8 − 4369 κ 6 + 819 κ 4 − 51 κ 2 + 1 ) − 3 {\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 1 − κ 2 ∏ n = 0 m + 1 [ 1 − ( 2 n − m − 1 ) κ ] m ! β m {\displaystyle {\frac {1-\kappa ^{2}}{\prod _{n=0}^{m+1}}}{\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 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} ={\frac {1-\kappa ^{2}}{\prod _{n=0}^{m+1}}}{\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} ={\frac {1}{\beta }}{\frac {1-\kappa ^{2}}{1-4\kappa ^{2}}}}

and the variance is:

Var ⁡ [ X ] = σ κ 2 = 1 β 2 2 ( 1 − 4 κ 2 ) 2 − ( 1 − κ 2 ) 2 ( 1 − 9 κ 2 ) ( 1 − 4 κ 2 ) 2 ( 1 − 9 κ 2 ) {\displaystyle \operatorname {Var} =\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:

Kurt ⁡ [ X ] = E ⁡ [ [ X − 1 β 1 − κ 2 1 − 4 κ 2 ] 4 σ κ 4 ] {\displaystyle \operatorname {Kurt} =\operatorname {E} \left^{4}}{\sigma _{\kappa }^{4}}}\right]}

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

Kurt ⁡ [ X ] = 9 ( 1 − κ 2 ) ( 1200 κ 14 − 6123 κ 12 + 562 κ 10 + 1539 κ 8 − 544 κ 6 + 143 κ 4 − 18 κ 2 + 1 ) β 4 σ κ 4 ( 1 − 4 κ 2 ) 4 ( 3600 κ 8 − 4369 κ 6 + 819 κ 4 − 51 κ 2 + 1 ) for 0 ≤ κ < 1 / 5 {\displaystyle \operatorname {Kurt} ={\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

Kurt ⁡ [ X ] = 9 ( 9 κ 2 − 1 ) 2 ( κ 2 − 1 ) ( 1200 κ 14 − 6123 κ 12 + 562 κ 10 + 1539 κ 8 − 544 κ 6 + 143 κ 4 − 18 κ 2 + 1 ) β 2 ( 1 − 4 κ 2 ) 2 ( 9 κ 6 + 13 κ 4 − 5 κ 2 + 1 ) ( 3600 κ 8 − 4369 κ 6 + 819 κ 4 − 51 κ 2 + 1 ) for 0 ≤ κ < 1 / 5 {\displaystyle \operatorname {Kurt} ={\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .

Skewness

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

Skew ⁡ [ X ] = E ⁡ [ [ X − 1 β 1 − κ 2 1 − 4 κ 2 ] 3 σ κ 3 ] {\displaystyle \operatorname {Skew} =\operatorname {E} \left^{3}}{\sigma _{\kappa }^{3}}}\right]}

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

Shew ⁡ [ X ] = 2 ( 1 − κ 2 ) ( 144 κ 8 + 23 κ 6 + 27 κ 4 − 6 κ 2 + 1 ) β 3 σ κ 3 ( 4 κ 2 − 1 ) 3 ( 144 κ 4 − 25 κ 2 + 1 ) for 0 ≤ κ < 1 / 4 {\displaystyle \operatorname {Shew} ={\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .

Type II

Probability density function

κ-exponential distribution of type II
Probability density function
Cumulative distribution function
Parameters 0 ≤ κ < 1 {\displaystyle 0\leq \kappa <1} shape (real)
β > 0 {\displaystyle \beta >0} rate (real)
Support x ∈ [ 0 , ∞ ) {\displaystyle x\in [0,\infty )}
PDF β 1 + κ 2 β 2 x 2 exp κ ⁡ ( − β x ) {\displaystyle {\frac {\beta }{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2}}}}\exp _{\kappa }(-\beta x)}
CDF 1 − exp k ⁡ ( − β x ) {\displaystyle 1-\exp _{k}({-\beta x)}}
Quantile β − 1 ln κ ⁡ ( 1 1 − F κ ) , 0 ≤ F κ ≤ 1 {\displaystyle \beta ^{-1}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )},0\leq F_{\kappa }\leq 1}
Mean 1 β 1 1 − κ 2 {\displaystyle {\frac {1}{\beta }}{\frac {1}{1-\kappa ^{2}}}}
Median β − 1 ln κ ⁡ ( 2 ) {\displaystyle \beta ^{-1}\ln _{\kappa }(2)}
Mode 1 κ β 2 ( 1 − κ 2 ) {\displaystyle {\frac {1}{\kappa \beta {\sqrt {2(1-\kappa ^{2})}}}}}
Variance σ κ 2 = 1 β 2 1 + 2 κ 4 ( 1 − 4 κ 2 ) ( 1 − κ 2 ) 2 {\displaystyle \sigma _{\kappa }^{2}={\frac {1}{\beta ^{2}}}{\frac {1+2\kappa ^{4}}{(1-4\kappa ^{2})(1-\kappa ^{2})^{2}}}}
Skewness 2 ( 15 κ 6 + 6 κ 4 + 2 κ 2 + 1 ) ( 1 − 9 κ 2 ) ( 2 κ 4 + 1 ) 1 − 4 κ 2 1 + 2 κ 4 {\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 3 ( 72 κ 10 − 360 κ 8 − 44 κ 6 − 32 κ 4 + 7 κ 2 − 3 ) ( 4 κ 2 − 1 ) − 1 ( 2 κ 4 + 1 ) 2 ( 144 κ 4 − 25 κ 2 + 1 ) {\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 β − m m ! ∏ n = 0 m [ 1 − ( 2 n − m ) κ ] {\displaystyle {\frac {\beta ^{-m}m!}{\prod _{n=0}^{m}}}}

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 α = 1 {\displaystyle \alpha =1} is:

f κ ( x ) = β 1 + κ 2 β 2 x 2 exp κ ⁡ ( − β x ) {\displaystyle f_{_{\kappa }}(x)={\frac {\beta }{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2}}}}\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 II is given by

F κ ( x ) = 1 − exp k ⁡ ( − β x ) {\displaystyle F_{\kappa }(x)=1-\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 II has moment of order m < 1 / κ {\displaystyle m<1/\kappa } given by

E ⁡ [ X m ] = β − m m ! ∏ n = 0 m [ 1 − ( 2 n − m ) κ ] {\displaystyle \operatorname {E} ={\frac {\beta ^{-m}m!}{\prod _{n=0}^{m}}}}

The expectation value and the variance are:

E ⁡ [ X ] = 1 β 1 1 − κ 2 {\displaystyle \operatorname {E} ={\frac {1}{\beta }}{\frac {1}{1-\kappa ^{2}}}}
Var ⁡ [ X ] = σ κ 2 = 1 β 2 1 + 2 κ 4 ( 1 − 4 κ 2 ) ( 1 − κ 2 ) 2 {\displaystyle \operatorname {Var} =\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 mode = 1 κ β 2 ( 1 − κ 2 ) {\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:

Kurt ⁡ [ X ] = E ⁡ [ ( X − 1 β 1 1 − κ 2 σ κ ) 4 ] {\displaystyle \operatorname {Kurt} =\operatorname {E} \left}

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

Kurt ⁡ [ X ] = 3 ( 72 κ 10 − 360 κ 8 − 44 κ 6 − 32 κ 4 + 7 κ 2 − 3 ) β 4 σ κ 4 ( κ 2 − 1 ) 4 ( 576 κ 6 − 244 κ 4 + 29 κ 2 − 1 )  for  0 ≤ κ < 1 / 4 {\displaystyle \operatorname {Kurt} ={\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

Kurt ⁡ [ X ] = 3 ( 72 κ 10 − 360 κ 8 − 44 κ 6 − 32 κ 4 + 7 κ 2 − 3 ) ( 4 κ 2 − 1 ) − 1 ( 2 κ 4 + 1 ) 2 ( 144 κ 4 − 25 κ 2 + 1 )  for  0 ≤ κ < 1 / 4 {\displaystyle \operatorname {Kurt} ={\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:

Skew ⁡ [ X ] = E ⁡ [ [ X − 1 β 1 1 − κ 2 ] 3 σ κ 3 ] {\displaystyle \operatorname {Skew} =\operatorname {E} \left^{3}}{\sigma _{\kappa }^{3}}}\right]}

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

Skew ⁡ [ X ] = − 2 ( 15 κ 6 + 6 κ 4 + 2 κ 2 + 1 ) β 3 σ κ 3 ( κ 2 − 1 ) 3 ( 36 κ 4 − 13 κ 2 + 1 ) for 0 ≤ κ < 1 / 3 {\displaystyle \operatorname {Skew} =-{\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

Skew ⁡ [ X ] = 2 ( 15 κ 6 + 6 κ 4 + 2 κ 2 + 1 ) ( 1 − 9 κ 2 ) ( 2 κ 4 + 1 ) 1 − 4 κ 2 1 + 2 κ 4 for 0 ≤ κ < 1 / 3 {\displaystyle \operatorname {Skew} ={\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 κ → 0 {\displaystyle \kappa \rightarrow 0} .

Quantiles

The quantiles are given by the following expression

x quantile ( F κ ) = β − 1 ln κ ⁡ ( 1 1 − F κ ) {\displaystyle x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}}

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

x median ( F κ ) = β − 1 ln κ ⁡ ( 2 ) {\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:

L κ ( F κ ) = 1 + 1 − κ 2 κ ( 1 − F κ ) 1 + κ − 1 + κ 2 κ ( 1 − F κ ) 1 − κ {\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

G κ = 2 + κ 2 4 − κ 2 {\displaystyle \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 → + ∞ f κ ( x ) ∼ κ − 1 ( 2 κ β ) − 1 / κ x ( − 1 − κ ) / κ {\displaystyle \lim _{x\to +\infty }f_{\kappa }(x)\sim \kappa ^{-1}(2\kappa \beta )^{-1/\kappa }x^{(-1-\kappa )/\kappa }}
lim x → 0 + f κ ( x ) = β {\displaystyle \lim _{x\to 0^{+}}f_{\kappa }(x)=\beta }

Applications

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

See also

References

  1. Kaniadakis, G. (2001). "Non-linear kinetics underlying generalized statistics". Physica A: Statistical Mechanics and Its Applications. 296 (3–4): 405–425. arXiv:cond-mat/0103467. Bibcode:2001PhyA..296..405K. doi:10.1016/S0378-4371(01)00184-4. S2CID 44275064.
  2. ^ Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.
  3. Oreste, Pierpaolo; Spagnoli, Giovanni (2018-04-03). "Statistical analysis of some main geomechanical formulations evaluated with the Kaniadakis exponential law". Geomechanics and Geoengineering. 13 (2): 139–145. doi:10.1080/17486025.2017.1373201. ISSN 1748-6025. S2CID 133860553.
  4. Ourabah, Kamel; Tribeche, Mouloud (2014). "Planck radiation law and Einstein coefficients reexamined in Kaniadakis κ statistics". Physical Review E. 89 (6): 062130. Bibcode:2014PhRvE..89f2130O. doi:10.1103/PhysRevE.89.062130. ISSN 1539-3755. PMID 25019747.
  5. da Silva, Sérgio Luiz E. F.; dos Santos Lima, Gustavo Z.; Volpe, Ernani V.; de Araújo, João M.; Corso, Gilberto (2021). "Robust approaches for inverse problems based on Tsallis and Kaniadakis generalised statistics". The European Physical Journal Plus. 136 (5): 518. Bibcode:2021EPJP..136..518D. doi:10.1140/epjp/s13360-021-01521-w. ISSN 2190-5444. S2CID 236575441.
  6. Macedo-Filho, A.; Moreira, D.A.; Silva, R.; da Silva, Luciano R. (2013). "Maximum entropy principle for Kaniadakis statistics and networks". Physics Letters A. 377 (12): 842–846. Bibcode:2013PhLA..377..842M. doi:10.1016/j.physleta.2013.01.032.

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