Kaniadakis logistic distribution

κ-Logistic distribution

Probability density function Plot of the κ-Logistic distribution for typical κ-values and [math]\displaystyle{ \beta = 1 }[/math]. The case [math]\displaystyle{ \kappa = 0 }[/math] corresponds to the ordinary Logistic distribution.
Cumulative distribution function Plots of the cumulative κ-Logistic distribution for typical κ-values and [math]\displaystyle{ \beta = 1 }[/math]. The case [math]\displaystyle{ \kappa = 0 }[/math] corresponds to the ordinary Logistic case.
Parameters	[math]\displaystyle{ 0 \leq \kappa \lt 1 }[/math] [math]\displaystyle{ \alpha \gt 0 }[/math] shape (real) [math]\displaystyle{ \beta\gt 0 }[/math] rate (real) [math]\displaystyle{ \lambda \gt 0 }[/math]
Support	[math]\displaystyle{ x \in [0, \infty) }[/math]
PDF	[math]\displaystyle{ \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 } }[/math]
CDF	[math]\displaystyle{ \frac{ 1 - \exp_\kappa(-\beta x^\alpha) }{ 1 + (\lambda - 1) \exp_\kappa(-\beta x^\alpha) } }[/math]

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 ([math]\displaystyle{ 0 \lt \lambda \lt 1 }[/math]) or fermionic ([math]\displaystyle{ \lambda \gt 1 }[/math]) 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:

[math]\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 } }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math], where [math]\displaystyle{ 0 \leq |\kappa| \lt 1 }[/math] is the entropic index associated with the Kaniadakis entropy, [math]\displaystyle{ \beta \gt 0 }[/math] is the rate parameter, [math]\displaystyle{ \lambda \gt 0 }[/math], and [math]\displaystyle{ \alpha \gt 0 }[/math] is the shape parameter.

The Logistic distribution is recovered as [math]\displaystyle{ \kappa \rightarrow 0. }[/math]

Cumulative distribution function

The cumulative distribution function of κ-Logistic is given by

[math]\displaystyle{ F_\kappa(x) = \frac{ 1 - \exp_\kappa(-\beta x^\alpha) }{ 1 + (\lambda - 1) \exp_\kappa(-\beta x^\alpha) } }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math]. The cumulative Logistic distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

Survival and hazard functions

The survival distribution function of κ-Logistic distribution is given by

[math]\displaystyle{ S_\kappa(x) = \frac{\lambda}{\exp_\kappa(\beta x^\alpha) + \lambda - 1} }[/math]

valid for [math]\displaystyle{ x \geq 0 }[/math]. The survival Logistic distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

The hazard function associated with the κ-Logistic distribution is obtained by the solution of the following evolution equation:

[math]\displaystyle{ \frac{ S_\kappa(x) }{ dx } = -h_\kappa S_\kappa(x) \left( 1 - \frac{ \lambda -1 }{ \lambda } S_\kappa(x) \right) }[/math]

with [math]\displaystyle{ S_\kappa(0) = 1 }[/math], where [math]\displaystyle{ h_\kappa }[/math] is the hazard function:

[math]\displaystyle{ h_\kappa = \frac{\alpha \beta x^{\alpha-1}}{\sqrt{1+\kappa^2 \beta^2 x^{2\alpha} }} }[/math]

The cumulative Kaniadakis κ-Logistic distribution is related to the hazard function by the following expression:

[math]\displaystyle{ S_\kappa = e^{-H_\kappa(x)} }[/math]

where [math]\displaystyle{ H_\kappa (x) = \int_0^x h_\kappa(z) dz }[/math] is the cumulative hazard function. The cumulative hazard function of the Logistic distribution is recovered in the classical limit [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

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 [math]\displaystyle{ \kappa \rightarrow 0 }[/math].
• The κ-Logistic distribution is a generalization of the κ-Weibull distribution when [math]\displaystyle{ \lambda = 1 }[/math].
• A κ-Logistic distribution corresponds to a Half-Logistic distribution when [math]\displaystyle{ \lambda = 2 }[/math], [math]\displaystyle{ \alpha = 1 }[/math] and [math]\displaystyle{ \kappa = 0 }[/math].
• The ordinary Logistic distribution is a particular case of a κ-Logistic distribution, when [math]\displaystyle{ \kappa = 0 }[/math].

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 [math]\displaystyle{ \kappa \rightarrow 0 }[/math].

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