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

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