Kaniadakis logistic distribution

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 ${\displaystyle \kappa \rightarrow 0.}$

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 ${\displaystyle \kappa \rightarrow 0}$.

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 ${\displaystyle \kappa \rightarrow 0}$.