| κ-Gamma distribution | |
| Type: | density |
| Parameters: | 0\leq\kappa<1 \alpha>0 \beta>0 0<\nu<1/\kappa |
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.
The Kaniadakis κ-Gamma distribution has the following probability density function:[1]
| f | |
| \kappa |
(x)=(1+\kappa\nu)(2\kappa)\nu
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| \alpha\beta\nu | |
| \Gamma(\nu) |
x\alpha\exp\kappa(-\betax\alpha)
valid for
x\geq0
0\leq|\kappa|<1
0<\nu<1/\kappa
\beta>0
\alpha>0
The ordinary generalized Gamma distribution is recovered as
\kappa → 0
| f | |
| 0 |
(x)=
| |\alpha|\beta\nu | |
| \Gamma\left(\nu\right) |
x\alpha\exp\kappa(-\betax\alpha)
The cumulative distribution function of κ-Gamma distribution assumes the form:
F\kappa(x)=(1+\kappa\nu)(2\kappa)\nu
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| \alpha\beta\nu | |
| \Gamma(\nu) |
| x | |
| \int | |
| 0 |
z\alpha\exp\kappa(-\betaz\alpha)dz
valid for
x\geq0
0\leq|\kappa|<1
\kappa → 0
The κ-Gamma distribution has moment of order
m
\operatorname{E}[Xm]=\beta-m/
| (1+\kappa\nu)(2\kappa)-m/\alpha | |||||
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| \Gamma(\nu) |
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The moment of order
m
0<\nu+m/\alpha<1/\kappa
The mode is given by:
xrm{mode
The κ-Gamma distribution behaves asymptotically as follows:
\limxf\kappa(x)\sim(2\kappa\beta)-1/\kappa(1+\kappa\nu)(2\kappa)\nu
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| \alpha\beta\nu | |
| \Gamma(\nu) |
x\alpha
| \lim | |
| x\to0+ |
f\kappa(x)=(1+\kappa\nu)(2\kappa)\nu
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| \alpha\beta\nu | |
| \Gamma(\nu) |
x\alpha
\alpha=\nu=1
\alpha=1
\nu=n=
\alpha=2
\nu=1/2
\kappa=0
\alpha=1
\alpha=\nu=1
\alpha=1
\nu=n=
\alpha=1
\nu=
\alpha=2
\nu>0
\alpha=2
\nu=1
\alpha=2
\nu=
\alpha=2
\nu=3/2
\alpha=2
\nu=1/2
\alpha>0
\nu=1
\alpha>0
\nu=1/\alpha