Kaniadakis logistic distribution

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 (${\displaystyle 0<\lambda <1}${\displaystyle 0<\lambda <1}) or fermionic (${\displaystyle \lambda >1}${\displaystyle \lambda >1}) character.^[1]

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]

${\displaystyle f_{_{\kappa }}(x)={\frac {\lambda \alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}{\frac {\exp _{\kappa }(-\beta x^{\alpha })}{[1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })]^{2}}}}${\displaystyle 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 ${\displaystyle x\geq 0}${\displaystyle x\geq 0}, where ${\displaystyle 0\leq |\kappa |<1}${\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy, ${\displaystyle \beta >0}${\displaystyle \beta >0} is the rate parameter, ${\displaystyle \lambda >0}${\displaystyle \lambda >0}, and ${\displaystyle \alpha >0}${\displaystyle \alpha >0} is the shape parameter.

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

The cumulative distribution function of κ-Logistic is given by

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

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

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

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

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

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

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

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

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

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

where ${\displaystyle H_{\kappa }(x)=\int _{0}^{x}h_{\kappa }(z)dz}${\displaystyle 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 ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}.

• 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 ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}.^[1]
• The κ-Logistic distribution is a generalization of the κ-Weibull distribution when ${\displaystyle \lambda =1}${\displaystyle \lambda =1}.
• A κ-Logistic distribution corresponds to a Half-Logistic distribution when ${\displaystyle \lambda =2}${\displaystyle \lambda =2}, ${\displaystyle \alpha =1}${\displaystyle \alpha =1} and ${\displaystyle \kappa =0}${\displaystyle \kappa =0}.
• The ordinary Logistic distribution is a particular case of a κ-Logistic distribution, when ${\displaystyle \kappa =0}${\displaystyle \kappa =0}.

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 ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}.^{[2] [3] [4]}

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Erlang distribution

• Kaniadakis Statistics on arXiv.org

• Probability distributions
• Mathematical and quantitative methods (economics)

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