Kaniadakis Gaussian distribution
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Probability density function | |||
Cumulative distribution function | |||
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The Kaniadakis Gaussian distribution (also known as κ-Gaussian distribution) is a probability distribution which arises as a generalization of the Gaussian distribution from the maximization of the Kaniadakis entropy under appropriated constraints. It is one example of a Kaniadakis κ-distribution. The κ-Gaussian distribution has been applied successfully for describing several complex systems in economy,[1] geophysics,[2] astrophysics, among many others.
The κ-Gaussian distribution is a particular case of the κ-Generalized Gamma distribution.[3]
Definitions
Probability density function
The general form of the centered Kaniadakis κ-Gaussian probability density function is:[3]
where is the entropic index associated with the Kaniadakis entropy, is the scale parameter, and
is the normalization constant.
The
Cumulative distribution function
The cumulative distribution function of κ-Gaussian distribution is given by
where
is the Kaniadakis κ-Error function, which is a generalization of the ordinary Error function as .
Properties
Moments, mean and variance
The centered κ-Gaussian distribution has a moment of odd order equal to zero, including the mean.
The variance is finite for and is given by:
Kurtosis
The kurtosis of the centered κ-Gaussian distribution may be computed thought:
which can be written as
Thus, the kurtosis of the centered κ-Gaussian distribution is given by:
or
κ-Error function
κ-Error function | |
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General information | |
General definition | |
Fields of application | Probability, thermodynamics |
Domain, codomain and image | |
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Derivative |
The Kaniadakis κ-Error function (or κ-Error function) is a one-parameter generalization of the ordinary error function defined as:[3]
Although the error function cannot be expressed in terms of elementary functions, numerical approximations are commonly employed.
For a random variable X distributed according to a κ-Gaussian distribution with mean 0 and standard deviation , κ-Error function means the probability that X falls in the interval .
Applications
The κ-Gaussian distribution has been applied in several areas, such as:
- In stock prices.[4]
- In inverse problems, Error laws in extreme statistics are robustly represented by κ-Gaussian distributions.[2][5][6]
- In astrophysics, stellar-residual-radial-velocity data have a Gaussian-type statistical distribution, in which the K index presents a strong relationship with the stellar-cluster ages.[7][8]
- In nuclear physics, the study of Doppler broadening function in nuclear reactors is well described by a κ-Gaussian distribution for analyzing the neutron-nuclei interaction.[9][10]
- In cosmology, for interpreting the dynamical evolution of the Friedmann–Robertson–Walker Universe.
- In Langmuir waves.[12]
See also
- Giorgio Kaniadakis
- Kaniadakis statistics
- Kaniadakis distribution
- Kaniadakis κ-Exponential distribution
- Kaniadakis κ-Gamma distribution
- Kaniadakis κ-Weibull distribution
- Kaniadakis κ-Logistic distribution
- Kaniadakis κ-Erlang distribution