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.

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 $$\nu = 1/\alpha $$;

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^2x^{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


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

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