# 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.

## 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 $\kappa \rightarrow 0$.