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

:$f_i=\exp_(-\beta E_i+\beta \mu)$

enables the power-law description of complex systems following the consistent κ-generalized statistical theory., where $\exp_(x)=(\sqrt+\kappa x)^$ 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 $\nu =$ 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-Exponential distribution

κ-Gaussian distribution

κ-Gamma distribution

κ-Weibull distribution

κ-Logistic distribution

κ-Erlang distribution

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:

: $f_(x) = \frac (2\kappa \beta )^ \left(1 - \frac \right) x^ \exp_\kappa(-\beta x^\alpha)$

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:

: $F_\kappa(x) = (2\kappa \beta )^ x^ \exp_\kappa(-\beta x^\alpha)$

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 ^m= \frac \frac

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

See also

* Giorgio Kaniadakis
* Kaniadakis statistics
* Kaniadakis κ-Exponential distribution
* Kaniadakis κ-Gaussian distribution
* Kaniadakis κ-Gamma distribution
* Kaniadakis κ-Weibull distribution
* Kaniadakis κ-Logistic distribution
* Kaniadakis κ-Erlang distribution

References

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External links

Giorgio Kaniadakis Google Scholar pageKaniadakis Statistics on arXiv.org

Probability distributions
Statistical mechanics