Kaniadakis Gamma distribution
The Kaniadakis Generalized Gamma distribution, 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
Probability density function
The Kaniadakis κ-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 κ-Gamma distribution assumes the form:valid for, where. The cumulative Generalized Gamma distribution is recovered in the classical limit.
Properties
Moments and mode
The κ-Gamma distribution has moment of order given byThe 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: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 ;