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.

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 $$\kappa \rightarrow 0$$.

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