Kaniadakis Gamma 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:

$f_{{}_{\kappa }}(x)=(1+\kappa \nu )(2\kappa {)}^{\nu }\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2})}\frac{\alpha {\beta }^{\nu }}{\mathrm{\Gamma }\left(\nu \right)}{x}^{\alpha \nu -1}{\exp }_{\kappa }(-\beta x^{\alpha })$

valid for $x\ge 0$, where $0\le |\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $0<\nu <1/\kappa$, $\beta >0$ is the scale parameter, and $\alpha >0$ is the shape parameter.

The ordinary generalized Gamma distribution is recovered as $\kappa \to 0$: $f_{{}_0}(x)=\frac{|\alpha |{\beta }^{\nu }}{\mathrm{\Gamma }\left(\nu \right)}{x}^{\alpha \nu -1}{\exp }_{\kappa }(-\beta x^{\alpha })$.

Cumulative distribution function

The cumulative distribution function of κ-Gamma distribution assumes the form:

$F_{\kappa }(x)=(1+\kappa \nu )(2\kappa {)}^{\nu }\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2})}\frac{\alpha {\beta }^{\nu }}{\mathrm{\Gamma }\left(\nu \right)}\underset{0}{\overset{x}{\int }}{z}^{\alpha \nu -1}{\exp }_{\kappa }(-\beta z^{\alpha })dz$

valid for $x\ge 0$, where $0\le |\kappa |<1$. The cumulative Generalized Gamma distribution is recovered in the classical limit $\kappa \to 0$.

Properties

Moments and mode

The κ-Gamma distribution has moment of order $m$ given by

$\mathrm{E}[X^m]={\beta }^{-m/\alpha }\frac{(1+\kappa \nu )(2\kappa {)}^{-m/\alpha }}{1+\kappa (\nu +\frac{m}{\alpha })}\frac{\mathrm{\Gamma }(\nu +\frac{m}{\alpha })}{\mathrm{\Gamma }(\nu )}\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2})}\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2}-\frac{m}{2\alpha })}{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2}+\frac{m}{2\alpha })}$

The moment of order $m$ of the κ-Gamma distribution is finite for $0<\nu +m/\alpha <1/\kappa$.

The mode is given by:

$x_{\text{mode}}={\beta }^{-1/\alpha }(\nu -\frac{1}{\alpha }{)}^{\frac{1}{\alpha }}[1-{\kappa }^2(\nu -\frac{1}{\alpha }{)}^2{]}^{-\frac{1}{2\alpha }}$

Asymptotic behavior

The κ-Gamma distribution behaves asymptotically as follows:

$\underset{x\to +\mathrm{\infty }}{\lim }{f}_{\kappa }(x)\sim (2\kappa \beta {)}^{-1/\kappa }(1+\kappa \nu )(2\kappa {)}^{\nu }\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2})}\frac{\alpha {\beta }^{\nu }}{\mathrm{\Gamma }\left(\nu \right)}{x}^{\alpha \nu -1-\alpha /\kappa }$

$\underset{x\to 0^{+}}{\lim }{f}_{\kappa }(x)=(1+\kappa \nu )(2\kappa {)}^{\nu }\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{\nu }{2})}{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{\nu }{2})}\frac{\alpha {\beta }^{\nu }}{\mathrm{\Gamma }\left(\nu \right)}{x}^{\alpha \nu -1}$

Related distributions

• The κ-Gamma distributions is a generalization of:
  • κ-Exponential distribution of type I, when $\alpha =\nu =1$;
  • Kaniadakis κ-Erlang distribution, when $\alpha =1$ and $\nu =n=$ positive integer.
  • κ-Half-Normal distribution, when $\alpha =2$ and $\nu =1/2$;
• A κ-Gamma distribution corresponds to several probability distributions when $\kappa =0$, such as:
  • Gamma distribution, when $\alpha =1$;
  • Exponential distribution, when $\alpha =\nu =1$;
  • Erlang distribution, when $\alpha =1$ and $\nu =n=$ positive integer;
  • Chi-Squared distribution, when $\alpha =1$ and $\nu =$ half integer;
  • Nakagami distribution, when $\alpha =2$ and $\nu >0$;
  • Rayleigh distribution, when $\alpha =2$ and $\nu =1$;
  • Chi distribution, when $\alpha =2$ and $\nu =$ half integer;
  • Maxwell distribution, when $\alpha =2$ and $\nu =3/2$;
  • Half-Normal distribution, when $\alpha =2$ and $\nu =1/2$;
  • Weibull distribution, when $\alpha >0$ and $\nu =1$;
  • Stretched Exponential distribution, when $\alpha >0$ and $\nu =1/\alpha$;

See also

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Weibull distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

External links

• Kaniadakis Statistics on arXiv.org

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