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

enables the power-law description of complex systems following the consistent κ-generalized statistical theory., where 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
• 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

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
• The Kaniadakis Gamma distribution, also called the κ-Gamma distribution, which is a four-parameter () deformation of the generalized Gamma distribution.
• The κ-Gamma distribution becomes a ...
• κ-Exponential distribution of Type I when .
• κ-Erlang distribution when and positive integer.
• κ-Half-Normal distribution, when and .
• Generalized Gamma distribution, when ;
• In the limit , the κ-Gamma distribution becomes a ...
• Erlang distribution, when and positive integer;
• Chi-Squared distribution, when and half integer;
• Nakagami distribution, when and ;
• Rayleigh distribution, when and ;
• Chi distribution, when and half integer;
• Maxwell distribution, when and ;
• Half-Normal distribution, when and ;
• Weibull distribution, when and ;
• Stretched Exponential distribution, when and ;

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:

valid for , where is the entropic index associated with the Kaniadakis entropy, is the scale parameter, and is the shape parameter.

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

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 .

Its moment of order given by

The moment of order of the κ-Distribution Type IV is finite for .

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

External links

• [https://scholar.google.it/citations?user=pFlYesUAAAAJ&hl=en Giorgio Kaniadakis Google Scholar page]
• Kaniadakis Statistics on arXiv.org