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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.
Definitions
[edit]Probability density function
[edit]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
[edit]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 .
Properties
[edit]Moments and mode
[edit]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
[edit]The κ-Gamma distribution behaves asymptotically as follows:[1]
Related distributions
[edit]- 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 ;
See also
[edit]- Giorgio Kaniadakis
- Kaniadakis statistics
- Kaniadakis distribution
- Kaniadakis κ-Exponential distribution
- Kaniadakis κ-Gaussian distribution
- Kaniadakis κ-Weibull distribution
- Kaniadakis κ-Logistic distribution
- Kaniadakis κ-Erlang distribution
References
[edit]- ^ a b c Kaniadakis, G. (2021-01-01). "New power-law tailed distributions emerging in κ-statistics (a)". Europhysics Letters. 133 (1): 10002. arXiv:2203.01743. Bibcode:2021EL....13310002K. doi:10.1209/0295-5075/133/10002. ISSN 0295-5075. S2CID 234144356.
External links
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