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.

κ-Gamma distribution

Probability density function
Parameters	$0\leq \kappa <1$ $\alpha >0$ shape (real) $\beta >0$ rate (real) $0<\nu <1/\kappa$
Support	$x\in [0,+\infty )$
PDF	$(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1}\exp _{\kappa }(-\beta x^{\alpha })$
CDF	$(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}\int _{0}^{x}z^{\alpha \nu -1}\exp _{\kappa }(-\beta z^{\alpha })dz$
Mode	$\beta ^{-1/\alpha }{\Bigg (}\nu -{\frac {1}{\alpha }}{\Bigg )}^{\frac {1}{\alpha }}{\Bigg [}1-\kappa ^{2}{\bigg (}\nu -{\frac {1}{\alpha }}{\bigg )}^{2}{\Bigg ]}^{-{\frac {1}{2\alpha }}}$
Method of Moments	$\beta ^{-m/\alpha }{\frac {(1+\kappa \nu )(2\kappa )^{-m/\alpha }}{1+\kappa {\big (}\nu +{\frac {m}{\alpha }}{\big )}}}{\frac {\Gamma {\big (}\nu +{\frac {m}{\alpha }}{\big )}}{\Gamma (\nu )}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\Big )}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}+{\frac {m}{2\alpha }}{\Big )}}}$

Definitions

Probability density function

The Kaniadakis κ-Gamma distribution has the following probability density function:

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

valid for $x\geq 0$, where $0\leq |\kappa |<1$ 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 $\kappa \rightarrow 0$: $f_{_{0}}(x)={\frac {|\alpha |\beta ^{\nu }}{\Gamma \left(\nu \right)}}x^{\alpha \nu -1}\exp _{\kappa }(-\beta x^{\alpha })$.

Cumulative distribution function

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

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

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

Properties

Moments and mode

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

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

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

The mode is given by:

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

Asymptotic behavior

The κ-Gamma distribution behaves asymptotically as follows:

$\lim _{x\to +\infty }f_{\kappa }(x)\sim (2\kappa \beta )^{-1/\kappa }(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1-\alpha /\kappa }$

$\lim _{x\to 0^{+}}f_{\kappa }(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big (}\nu {\big )}}}x^{\alpha \nu -1}$

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