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 < λ < 1 {\displaystyle 0<\lambda <1} ) or fermionic ( λ > 1 {\displaystyle \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 κ ( x ) = λ α β x α − 1 1 + κ 2 β 2 x 2 α exp κ ( − β x α ) [ 1 + ( λ − 1 ) exp κ ( − β x α ) ] 2 {\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 x ≥ 0 {\displaystyle x\geq 0} , where 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy, β > 0 {\displaystyle \beta >0} is the rate parameter, λ > 0 {\displaystyle \lambda >0} , and α > 0 {\displaystyle \alpha >0} is the shape parameter. The Logistic distribution is recovered as κ → 0. {\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function The cumulative distribution function of κ-Logistic is given by
F κ ( x ) = 1 − exp κ ( − β x α ) 1 + ( λ − 1 ) exp κ ( − β x α ) {\displaystyle F_{\kappa }(x)={\frac {1-\exp _{\kappa }(-\beta x^{\alpha })}{1+(\lambda -1)\exp _{\kappa }(-\beta x^{\alpha })}}}
valid for x ≥ 0 {\displaystyle x\geq 0} . The cumulative Logistic distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
Survival and hazard functions The survival distribution function of κ-Logistic distribution is given by
S κ ( x ) = λ exp κ ( β x α ) + λ − 1 {\displaystyle S_{\kappa }(x)={\frac {\lambda }{\exp _{\kappa }(\beta x^{\alpha })+\lambda -1}}}
valid for x ≥ 0 {\displaystyle x\geq 0} . The survival Logistic distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
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