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 ($0<\lambda <1$) or fermionic ($\lambda >1$) character.

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

Definitions

Probability density function

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:

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

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

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

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

• 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 $\kappa \rightarrow 0$.
• 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

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 $\kappa \rightarrow 0$.

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