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 Giorgio Kaniadakis Google Scholar page Kaniadakis Statistics on arXiv.org