Kaniadakis distribution
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In
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.,[16][17] 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 [18]
Supported on semi-infinite intervals, usually [0,∞)
- The Kaniadakis Exponential distribution, also called the κ-Exponential distribution. The exponential distributionis 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 distributionof 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 ;
- The κ-Gamma distribution becomes a ...
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
Probability density function | |||
Cumulative distribution function | |||
Parameters |
shape (real) rate (real) | ||
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Support | |||
CDF | |||
Method of Moments |
The Kaniadakis distribution of Type IV (or κ-Distribution Type IV) is a three-parameter family of continuous statistical distributions.[1]
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
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
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