The Kaniadakis Generalized Gamma distribution (or ú-Generalized Gamma distribution) is a four-parameter family of continuous statistical distributions, supported on a semi-infinite interval [0,âÂÂ), which arising from the Kaniadakis statistics. It is one example of a Kaniadakis distribution. The ú-Gamma is a deformation of the Generalized Gamma distribution.
Definitions
Probability density function
The Kaniadakis ú-Gamma 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 ordinary generalized Gamma distribution is recovered as : .
Cumulative distribution function
The cumulative distribution function of ú-Gamma distribution assumes the form:
valid for , where . The cumulative Generalized Gamma distribution is recovered in the classical limit .
Properties
Moments and mode
The ú-Gamma distribution has moment of order given by
The moment of order of the ú-Gamma distribution is finite for .
The mode is given by:
Asymptotic behavior
The ú-Gamma distribution behaves asymptotically as follows:
Related distributions
- The ú-Gamma distributions is a generalization of:
- ú-Exponential distribution of type I, when ;
- Kaniadakis ú-Erlang distribution, when and positive integer.
- ú-Half-Normal distribution, when and ;
- A ú-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
References
External links
- [https://arxiv.org/search/?query=kaniadakis+statistics&searchtype=all&abstracts=show&order=-announced_date_first&size=200 Kaniadakis Statistics on arXiv.org]