# Kaniadakis distribution

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
    $f_{i}=\exp _{\kappa }(-\beta E_{i}+\beta \mu )$
enables the power-law description of complex systems following the consistent κ-generalized statistical theory., where $\exp _{\kappa }(x)=({\sqrt {1+\kappa ^{2}x^{2}}}+\kappa x)^{1/\kappa }$ 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 $\kappa \rightarrow 0.$
* 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 $\kappa \rightarrow 0.$

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

## Common Kaniadakis distributions
### κ-Distribution Type IV

The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.
The κ-Distribution Type IV distribution has the following probability density function:
    $f_{_{\kappa }}(x)={\frac {\alpha }{\kappa }}(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha }}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })$
valid for $x\geq 0$, where $0\leq |\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $\beta >0$ is the scale parameter, and $\alpha >0$ is the shape parameter.
The cumulative distribution function of κ-Distribution Type IV assumes the form:
    $F_{\kappa }(x)=(2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })$
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 $\kappa \rightarrow 0$.
Its moment of order $m$ given by
    $\operatorname {E} [X^{m}]={\frac {(2\kappa \beta )^{-m/\alpha }}{1+\kappa {\frac {m}{2\alpha }}}}{\frac {\Gamma {\Big{\frac {1}{\kappa }}+{\frac {m}{\alpha }}{\Big )}\Gamma {\Big1-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big{\frac {1}{\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}$
The moment of order $m$ of the κ-Distribution Type IV is finite for $m<2\alpha$.