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κ-Logistic distribution
	Probability density function Plot of the κ-Logistic distribution for typical κ-values and . The case corresponds to the ordinary Logistic distribution.
	Cumulative distribution function Plots of the cumulative κ-Logistic distribution for typical κ-values and . The case corresponds to the ordinary Logistic case.
Parameters	; shape (real) ; rate (real) ;
Support
PDF
CDF

The Kaniadakis Logistic distribution (also known as κ-Logisticdistribution) is a generalized version of the Logistic distribution associated with the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The κ-Logistic probability distribution describes the population kinetics behavior of bosonic ( $0<\lambda <1$ ) or fermionic ( $\lambda >1$ ) character.^[1]

Definitions[edit]

Probability density function[edit]

The Kaniadakis κ-Logistic distribution is a four-parameter family of continuous statistical distributions, which is part of a class of statistical distributions emerging from the Kaniadakis κ-statistics. This distribution has the following probability density function:^[1]

f_{_{\kappa }}(x)={\frac {\lambda \alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}{\frac {\exp _{\kappa }(-\beta x^{\alpha })}{[1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })]^{2}}}

valid for $x\geq 0$ , where $0\leq |\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $\beta >0$ is the rate parameter, $\lambda >0$ , and $\alpha >0$ is the shape parameter.

The Logistic distribution is recovered as $\kappa \rightarrow 0.$

Cumulative distribution function[edit]

The cumulative distribution function of κ-Logistic is given by

F_{\kappa }(x)={\frac {1-\exp _{\kappa }(-\beta x^{\alpha })}{1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })}}

valid for $x\geq 0$ . The cumulative Logistic distribution is recovered in the classical limit $\kappa \rightarrow 0$ .

Survival and hazard functions[edit]

The survival distribution function of κ-Logistic distribution is given by

S_{\kappa }(x)={\frac {\lambda }{\exp _{\kappa }(\beta x^{\alpha })+\lambda -1}}

valid for $x\geq 0$ . The survival Logistic distribution is recovered in the classical limit $\kappa \rightarrow 0$ .

The hazard function associated with the κ-Logistic distribution is obtained by the solution of the following evolution equation:

${\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)\left(1-{\frac {\lambda -1}{\lambda }}S_{\kappa }(x)\right)$

with $S_{\kappa }(0)=1$ , where $h_{\kappa }$ is the hazard function:

h_{\kappa }={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}

The cumulative Kaniadakis κ-Logistic distribution is related to the hazard function by the following expression:

S_{\kappa }=e^{-H_{\kappa }(x)}

where $H_{\kappa }(x)=\int _{0}^{x}h_{\kappa }(z)dz$ is the cumulative hazard function. The cumulative hazard function of the Logistic distribution is recovered in the classical limit $\kappa \rightarrow 0$ .

Related distributions[edit]

The survival function of the κ-Logistic distribution represents the κ-deformation of the Fermi-Dirac function, and becomes a Fermi-Dirac distribution in the classical limit $\kappa \rightarrow 0$ .^[1]
The κ-Logistic distribution is a generalization of the κ-Weibull distribution when $\lambda =1$ .
A κ-Logistic distribution corresponds to a Half-Logistic distribution when $\lambda =2$ , $\alpha =1$ and $\kappa =0$ .
The ordinary Logistic distribution is a particular case of a κ-Logistic distribution, when $\kappa =0$ .

Applications[edit]

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

In quantum statistics, the survival function of the κ-Logistic distribution represents the most general expression of the Fermi-Dirac function, reducing to the Fermi-Dirac distribution in the limit $\kappa \rightarrow 0$ .^[2]^[3]^[4]

References[edit]

^ ^a ^b ^c 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.
^ Santos, A.P.; Silva, R.; Alcaniz, J.S.; Anselmo, D.H.A.L. (2011). "Kaniadakis statistics and the quantum H-theorem". Physics Letters A. 375 (3): 352–355. Bibcode:2011PhLA..375..352S. doi:10.1016/j.physleta.2010.11.045.
^ Kaniadakis, G. (2001). "H-theorem and generalized entropies within the framework of nonlinear kinetics". Physics Letters A. 288 (5–6): 283–291. arXiv:cond-mat/0109192. Bibcode:2001PhLA..288..283K. doi:10.1016/S0375-9601(01)00543-6. S2CID 119445915.
^ Lourek, Imene; Tribeche, Mouloud (2017). "Thermodynamic properties of the blackbody radiation: A Kaniadakis approach". Physics Letters A. 381 (5): 452–456. Bibcode:2017PhLA..381..452L. doi:10.1016/j.physleta.2016.12.019.

External links[edit]

Kaniadakis Statistics on arXiv.org

[:0-1] 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.

[2] Santos, A.P.; Silva, R.; Alcaniz, J.S.; Anselmo, D.H.A.L. (2011). "Kaniadakis statistics and the quantum H-theorem". Physics Letters A. 375 (3): 352–355. Bibcode:2011PhLA..375..352S. doi:10.1016/j.physleta.2010.11.045.

[3] Kaniadakis, G. (2001). "H-theorem and generalized entropies within the framework of nonlinear kinetics". Physics Letters A. 288 (5–6): 283–291. arXiv:cond-mat/0109192. Bibcode:2001PhLA..288..283K. doi:10.1016/S0375-9601(01)00543-6. S2CID 119445915.

[4] Lourek, Imene; Tribeche, Mouloud (2017). "Thermodynamic properties of the blackbody radiation: A Kaniadakis approach". Physics Letters A. 381 (5): 452–456. Bibcode:2017PhLA..381..452L. doi:10.1016/j.physleta.2016.12.019.

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Probability density function Plot of the κ-Logistic distribution for typical κ-values and $\beta =1$ . The case $\kappa =0$ corresponds to the ordinary Logistic distribution.
Cumulative distribution function Plots of the cumulative κ-Logistic distribution for typical κ-values and $\beta =1$ . The case $\kappa =0$ corresponds to the ordinary Logistic case.
Parameters	$0\leq \kappa <1$ $\alpha >0$ shape (real) $\beta >0$ rate (real) $\lambda >0$
Support	$x\in [0,\infty )$
PDF	${\frac {\lambda \alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}{\frac {\exp _{\kappa }(-\beta x^{\alpha })}{[1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })]^{2}}}$
CDF	${\frac {1-\exp _{\kappa }(-\beta x^{\alpha })}{1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })}}$

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