Kaniadakis distribution

Not to be confused with the K-distribution of probability distributions.

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

Plot of the κ-Gaussian distribution for typical κ-values. The case κ=0 corresponds to the normal distribution.

• 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,∞)

Plot of the κ-Gamma distribution for typical κ-values.

• 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-Exponential distribution

Main article: Kaniadakis Exponential distribution

κ-Gaussian distribution

Main article: Kaniadakis Gaussian distribution

κ-Gamma distribution

Main article: Kaniadakis Gamma distribution

κ-Weibull distribution

Main article: Kaniadakis Weibull distribution

κ-Logistic distribution

Main article: Kaniadakis Logistic distribution

κ-Erlang distribution

Main article: Kaniadakis 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 $\alpha =\beta =1$.
Cumulative distribution function
Parameters	$0\leq \kappa <1$ $\alpha >0$ shape (real) $\beta >0$ rate (real)
Support	$x\in [0,+\infty )$
PDF	${\frac {\alpha }{\kappa }}(2\kappa \beta )^{1/\kappa }\left(1-{\frac {\kappa \beta x^{\alpha }}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\right)x^{-1+\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })$
CDF	$(2\kappa \beta )^{1/\kappa }x^{\alpha /\kappa }\exp _{\kappa }(-\beta x^{\alpha })$
Method of moments	${\frac {(2\kappa \beta )^{-m/\alpha }}{1+\kappa {\frac {m}{2\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{\alpha }}{\Big )}\Gamma {\Big (}1-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}$

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

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

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

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

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

External links

• Giorgio Kaniadakis Google Scholar page
• Kaniadakis Statistics on arXiv.org