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Kaniadakis Gamma distribution

Continuous probability distribution From Wikipedia, the free encyclopedia

Kaniadakis Gamma distribution
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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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  • The κ-Gamma distributions is a generalization of:
  • 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

References

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