# 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:
    $f_{_{\kappa }}(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\big{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\big )}}{\Gamma {\big{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\big )}}}{\frac {\alpha \beta ^{\nu }}{\Gamma {\big\nu {\big )}}}x^{\alpha \nu -1}\exp _{\kappa }(-\beta x^{\alpha })$
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:
    $x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg\nu -{\frac {1}{\alpha }}{\Bigg )}^{\frac {1}{\alpha }}{\Bigg1-\kappa ^{2}{\bigg\nu -{\frac {1}{\alpha }}{\bigg )}^{2}{\Bigg ]}^{-{\frac {1}{2\alpha }}}$

### 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 $\nu =1/\alpha$;