Kaniadakis distribution
In statistics, a Kaniadakis 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-Distribution Type IV
The Kaniadakis distribution of 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.