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Continuous probability distribution
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
Probability density function
The Kaniadakis κ-Gamma distribution has the following
probability density function:
[1]
![{\displaystyle f_{_{\kappa }}(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1}\exp _{\kappa }(-\beta x^{\alpha })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbe2a285faf9f50b090090b20478ac9f2f2e7bf)
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:
![{\displaystyle F_{\kappa }(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}\int _{0}^{x}z^{\alpha \nu -1}\exp _{\kappa }(-\beta z^{\alpha })dz}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fa9dbf11a190373617efa428425e4ae6b4c4d6a)
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 by
[1]
![{\displaystyle \operatorname {E} [X^{m}]=\beta ^{-m/\alpha }{\frac {(1+\kappa \nu )(2\kappa )^{-m/\alpha }}{1+\kappa {\big (}\nu +{\frac {m}{\alpha }}{\big )}}}{\frac {\Gamma {\big (}\nu +{\frac {m}{\alpha }}{\big )}}{\Gamma (\nu )}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\Big )}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}+{\frac {m}{2\alpha }}{\Big )}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35dd73003c94f630463fb7ab4d77810cbe817f93)
The moment of order
of the κ-Gamma distribution is finite for
.
The mode is given by:
![{\displaystyle x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg (}\nu -{\frac {1}{\alpha }}{\Bigg )}^{\frac {1}{\alpha }}{\Bigg [}1-\kappa ^{2}{\bigg (}\nu -{\frac {1}{\alpha }}{\bigg )}^{2}{\Bigg ]}^{-{\frac {1}{2\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d8ce7ccd11ecd49e8d0aa04d870513e36952129)
Asymptotic behavior
The κ-Gamma distribution behaves
asymptotically as follows:
[1]
![{\displaystyle \lim _{x\to +\infty }f_{\kappa }(x)\sim (2\kappa \beta )^{-1/\kappa }(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1-\alpha /\kappa }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b47357d94e3dc6fa0d15d62815ce2dca6242e84)
![{\displaystyle \lim _{x\to 0^{+}}f_{\kappa }(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641292e2e576ddd838e3f6d6ca048804fb309347)
Related distributions
- 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
;
See also
References
External links
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Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and
singular | |
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Families | |
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