In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics. There are several families of Kaniadakis distributions related to different constraints used in the maximization of the Kaniadakis entropy, such as the κ-Exponential distribution, κ-Gaussian distribution, Kaniadakis κ-Gamma distribution and κ-Weibull distribution. The κ-distributions have been applied for modeling a vast phenomenology of experimental statistical distributions in natural or artificial complex systems, such as, in epidemiology, quantum statistics, in astrophysics and cosmology, in geophysics, in economy, in machine learning. The κ-distributions are written as function of the κ-deformed exponential, taking the form
f i = exp κ ( − β E i + β μ ) {\displaystyle f_{i}=\exp _{\kappa }(-\beta E_{i}+\beta \mu )}
enables the power-law description of complex systems following the consistent κ-generalized statistical theory., where exp κ ( x ) = ( 1 + κ 2 x 2 + κ x ) 1 / κ {\displaystyle \exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x)^{1/\kappa }} is the Kaniadakis κ-exponential function. The κ-distribution becomes the common Boltzmann distribution at low energies, while it has a power-law tail at high energies, the feature of high interest of many researchers.
List of κ-statistical distributions
Supported on the whole real line
The Kaniadakis Gaussian distribution, also called the κ-Gaussian distribution. The normal distribution is a particular case when κ → 0. {\displaystyle \kappa \rightarrow 0.}
The Kaniadakis double exponential distribution, as known as Kaniadakis κ-double exponential distribution or κ-Laplace distribution. The Laplace distribution is a particular case when κ → 0. {\displaystyle \kappa \rightarrow 0.}
Supported on semi-infinite intervals, usually [0,∞)
The Kaniadakis Exponential distribution, also called the κ-Exponential distribution. The exponential distribution is a particular case when κ → 0. {\displaystyle \kappa \rightarrow 0.}
The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter ( κ , α , β , ν {\displaystyle \kappa ,\alpha ,\beta ,\nu } ) deformation of the generalized Gamma distribution. The κ-Gamma distribution becomes a ... κ-Exponential distribution of Type I when α = ν = 1 {\displaystyle \alpha =\nu =1} . κ-Erlang distribution when α = 1 {\displaystyle \alpha =1} and ν = n = {\displaystyle \nu =n=} positive integer. κ-Half-Normal distribution, when α = 2 {\displaystyle \alpha =2} and ν = 1 / 2 {\displaystyle \nu =1/2} . Generalized Gamma distribution, when α = 1 {\displaystyle \alpha =1} ; In the limit κ → 0 {\displaystyle \kappa \rightarrow 0} , the κ-Gamma distribution becomes a ... Erlang distribution, when α = 1 {\displaystyle \alpha =1} and ν = n = {\displaystyle \nu =n=} positive integer; Chi-Squared distribution, when α = 1 {\displaystyle \alpha =1} and ν = {\displaystyle \nu =} half integer; Nakagami distribution, when α = 2 {\displaystyle \alpha =2} and ν > 0 {\displaystyle \nu >0} ; Rayleigh distribution, when α = 2 {\displaystyle \alpha =2} and ν = 1 {\displaystyle \nu =1} ; Chi distribution, when α = 2 {\displaystyle \alpha =2} and ν = {\displaystyle \nu =} half integer; Maxwell distribution, when α = 2 {\displaystyle \alpha =2} and ν = 3 / 2 {\displaystyle \nu =3/2} ; Half-Normal distribution, when α = 2 {\displaystyle \alpha =2} and ν = 1 / 2 {\displaystyle \nu =1/2} ; Weibull distribution, when α > 0 {\displaystyle \alpha >0} and ν = 1 {\displaystyle \nu =1} ; Stretched Exponential distribution, when α > 0 {\displaystyle \alpha >0} and ν = 1 / α {\displaystyle \nu =1/\alpha } ;
Common Kaniadakis distributions
κ-Exponential distribution
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