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.

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 $$\kappa \rightarrow 0$$.