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Kaniadakis Gamma distribution
Continuous probability distribution From Wikipedia, the free encyclopedia
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The Kaniadakis Generalized Gamma distribution (or κ-Generalized Gamma distribution) is a four-parameter family of continuous statistical distributions, supported on a semi-infinite interval [0,∞), which arising from the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The κ-Gamma is a deformation of the Generalized Gamma distribution.
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Definitions
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Probability density function
The Kaniadakis κ-Gamma distribution has the following probability density function:[1]
valid for , where is the entropic index associated with the Kaniadakis entropy, , is the scale parameter, and is the shape parameter.
The ordinary generalized Gamma distribution is recovered as : .
Cumulative distribution function
The cumulative distribution function of κ-Gamma distribution assumes the form:
valid for , where . The cumulative Generalized Gamma distribution is recovered in the classical limit .
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Properties
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Moments and mode
The κ-Gamma distribution has moment of order given by[1]
The moment of order of the κ-Gamma distribution is finite for .
The mode is given by:
Asymptotic behavior
The κ-Gamma distribution behaves asymptotically as follows:[1]
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Related distributions
- The κ-Gamma distributions is a generalization of:
- κ-Exponential distribution of type I, when ;
- Kaniadakis κ-Erlang distribution, when and positive integer.
- κ-Half-Normal distribution, when and ;
- A κ-Gamma distribution corresponds to several probability distributions when , such as:
- Gamma distribution, when ;
- Exponential distribution, when ;
- 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 ;
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See also
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