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κ -Logistic distribution

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

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 ( ${\displaystyle 0<\lambda <1}$) or fermionic ( ${\displaystyle \lambda >1}$) 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]

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

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

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

Cumulative distribution function [ edit ]

The cumulative distribution function of κ -Logistic is given by

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

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

Survival and hazard functions [ edit ]

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

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

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

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

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

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

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

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

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

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

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

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

See also [ edit ]

• 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

References [ edit ]

External links [ edit ]

• Kaniadakis Statistics on arXiv.org