Kaniadakis Gamma Distribution

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Definitions

Probability density function

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

: $f_(x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac \frac x^ \exp_\kappa(-\beta x^\alpha)$

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 $\kappa \rightarrow 0$: $f_(x) = \frac x^ \exp_\kappa(-\beta x^\alpha)$.

Cumulative distribution function

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

: $F_\kappa(x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac \frac \int_0^x z^ \exp_\kappa(-\beta z^\alpha) dz$

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 ^m= \beta^ \frac \frac \frac \frac

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_ = \beta^ \Bigg( \nu - \frac \Bigg)^ \Bigg 1 - \kappa^2 \bigg( \nu - \frac\bigg)^2\Bigg

Asymptotic behavior

The ''κ''-Gamma distribution behaves asymptotically as follows:

:$\lim_ f_\kappa (x) \sim (2\kappa \beta)^ (1 + \kappa \nu) (2 \kappa)^\nu \frac \frac x^$
:$\lim_ f_\kappa (x) = (1 + \kappa \nu) (2 \kappa)^\nu \frac \frac x^$

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

References

External links

Kaniadakis Statistics on arXiv.org

{{DEFAULTSORT:Kaniadakis Gamma Distribution
Statistics
Probability distributions
Continuous distributions
Survival analysis
Exponential family distributions
Infinitely divisible probability distributions