Kaniadakis logistic distribution

κ-Logistic distribution

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

TheKaniadakis Logistic distribution(also known asκ-Logisticdistribution)is a generalized version of theLogistic distributionassociated with theKaniadakis statistics.It is one example of aKaniadakis distribution.The κ-Logistic probability distribution describes the population kinetics behavior of bosonic ($0<\lambda <1$) or fermionic ($\lambda >1$) character.

Definitions

Probability density function

The Kaniadakisκ-Logistic distribution is a four-parameter family ofcontinuous statistical distributions,which is part of a class ofstatistical distributionsemerging from theKaniadakis κ-statistics.This distribution has the followingprobability density function:

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

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

TheLogistic distributionis recovered as$\kappa \rightarrow 0.$

Cumulative distribution function

Thecumulative distribution functionofκ-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

The survival distribution function ofκ-Logistic distribution is given by

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

valid for$x\geq 0$.The survivalLogistic distributionis recovered in the classical limit$\kappa \rightarrow 0$.

The hazard function associated with theκ-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 Kaniadakisκ-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 theLogistic distributionis recovered in the classical limit$\kappa \rightarrow 0$.

Related distributions

• The survival function of theκ-Logistic distribution represents theκ-deformation of the Fermi-Dirac function, and becomes aFermi-Dirac distributionin the classical limit$\kappa \rightarrow 0$.
• Theκ-Logistic distribution is a generalization of theκ-Weibull distributionwhen$\lambda =1$.
• Aκ-Logistic distribution corresponds to aHalf-Logistic distributionwhen$\lambda =2$,$\alpha =1$and$\kappa =0$.
• The ordinary Logistic distribution is a particular case of aκ-Logistic distribution, when$\kappa =0$.

Applications

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

• Inquantum statistics,the survival function of theκ-Logistic distribution represents the most general expression of the Fermi-Dirac function, reducing to theFermi-Dirac distributionin the limit$\kappa \rightarrow 0$.

See also

• 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

External links

• Kaniadakis Statistics on arXiv.org