In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics.^[1] 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,^[2] quantum statistics,^[3][4][5] in astrophysics and cosmology,^[6][7][8] in geophysics,^[9][10][11] in economy,^[12][13][14] in machine learning.^[15]

The κ-distributions are written as function of the κ-deformed exponential, taking the form

${\displaystyle f_{i}=\exp _{\kappa }(-\beta E_{i}+\beta \mu )}$

enables the power-law description of complex systems following the consistent κ-generalized statistical theory.,^[16][17] where ${\displaystyle \exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2))}+\kappa x)^{1/\kappa ))$ 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 ${\displaystyle \kappa \rightarrow 0.}$
• 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 ${\displaystyle \kappa \rightarrow 0.}$^[18]

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 ${\displaystyle \kappa \rightarrow 0.}$
• The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter (${\displaystyle \kappa ,\alpha ,\beta ,\nu }$) deformation of the generalized Gamma distribution.
  • The κ-Gamma distribution becomes a ...
    • κ-Exponential distribution of Type I when ${\displaystyle \alpha =\nu =1}$.
    • κ-Erlang distribution when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =n=}$ positive integer.
    • κ-Half-Normal distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1/2}$.
    • Generalized Gamma distribution, when ${\displaystyle \alpha =1}$;
  • In the limit ${\displaystyle \kappa \rightarrow 0}$, the κ-Gamma distribution becomes a ...
    • Erlang distribution, when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =n=}$ positive integer;
    • Chi-Squared distribution, when ${\displaystyle \alpha =1}$ and ${\displaystyle \nu =}$ half integer;
    • Nakagami distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu >0}$;
    • Rayleigh distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1}$;
    • Chi distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =}$ half integer;
    • Maxwell distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =3/2}$;
    • Half-Normal distribution, when ${\displaystyle \alpha =2}$ and ${\displaystyle \nu =1/2}$;
    • Weibull distribution, when ${\displaystyle \alpha >0}$ and ${\displaystyle \nu =1}$;
    • Stretched Exponential distribution, when ${\displaystyle \alpha >0}$ and ${\displaystyle \nu =1/\alpha }$;

Common Kaniadakis distributions

κ-Exponential distribution

κ-Gaussian distribution

κ-Gamma distribution

κ-Weibull distribution

κ-Logistic distribution

κ-Erlang distribution

κ-Distribution Type IV

κ-Distribution Type IV

Probability density function Plot of the κ-Distribution Type IV for typical κ-values, and ${\displaystyle \alpha =\beta =1}$.
Cumulative distribution function
Parameters	${\displaystyle 0\leq \kappa <1}$ ${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ rate (real)
Support	${\displaystyle x\in [0,+\infty )}$
PDF	${\displaystyle {\frac {\alpha }{\kappa ))(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha )){\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha ))))\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$
CDF	${\displaystyle (2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$
Method of Moments	${\displaystyle {\frac {(2\kappa \beta )^{-m/\alpha )){1+\kappa {\frac {m}{2\alpha )))){\frac {\Gamma {\Big (}{\frac {1}{\kappa ))+{\frac {m}{\alpha )){\Big )}\Gamma {\Big (}1-{\frac {m}{2\alpha )){\Big ))){\Gamma {\Big (}{\frac {1}{\kappa ))+{\frac {m}{2\alpha )){\Big )))))$

The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.^[1]

The κ-Distribution Type IV distribution has the following probability density function:

${\displaystyle f_{_{\kappa ))(x)={\frac {\alpha }{\kappa ))(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha )){\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha ))))\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$

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 scale parameter, and ${\displaystyle \alpha >0}$ is the shape parameter.

The cumulative distribution function of κ-Distribution Type IV assumes the form:

${\displaystyle F_{\kappa }(x)=(2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })}$

The κ-Distribution Type IV does not admit a classical version, since the probability function and its cumulative reduces to zero in the classical limit ${\displaystyle \kappa \rightarrow 0}$.

Its moment of order ${\displaystyle m}$ given by

${\displaystyle \operatorname {E} [X^{m}]={\frac {(2\kappa \beta )^{-m/\alpha )){1+\kappa {\frac {m}{2\alpha )))){\frac {\Gamma {\Big (}{\frac {1}{\kappa ))+{\frac {m}{\alpha )){\Big )}\Gamma {\Big (}1-{\frac {m}{2\alpha )){\Big ))){\Gamma {\Big (}{\frac {1}{\kappa ))+{\frac {m}{2\alpha )){\Big )))))$

The moment of order ${\displaystyle m}$ of the κ-Distribution Type IV is finite for ${\displaystyle m<2\alpha }$.