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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 ,∞), which arising from the Kaniadakis statistics. It is one example of a Kaniadakis distribution">Kaniadakis_statistics.html" ;"title=",∞), which arising from the Kaniadakis statistics">,∞), 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">Generalized Gamma distribution The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many di ...
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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 The generalized gamma distribution is a continuous probability distribution with two shape parameters (and a scale parameter). It is a generalization of the gamma distribution which has one shape parameter (and a scale parameter). Since many di ...
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
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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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
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 In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma dis ...
, when \alpha = 1; **
Exponential distribution In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
, when \alpha = \nu = 1; ** Erlang distribution, when \alpha = 1 and \nu = n = positive integer; **
Chi-Squared distribution In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-squar ...
, when \alpha = 1 and \nu = half integer; **
Nakagami distribution The Nakagami distribution or the Nakagami-''m'' distribution is a probability distribution related to the gamma distribution. The family of Nakagami distributions has two parameters: a shape parameter m\geq 1/2 and a second parameter controlling ...
, when \alpha = 2 and \nu > 0 ; **
Rayleigh distribution In probability theory and statistics, the Rayleigh distribution is a continuous probability distribution for nonnegative-valued random variables. Up to rescaling, it coincides with the chi distribution with two degrees of freedom. The distribu ...
, when \alpha = 2 and \nu = 1 ; **
Chi distribution In probability theory and statistics, the chi distribution is a continuous probability distribution. It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard no ...
, when \alpha = 2 and \nu = half integer; **Maxwell distribution, when \alpha = 2 and \nu = 3/2 ; **
Half-Normal distribution In probability theory and statistics, the half-normal distribution is a special case of the folded normal distribution. Let X follow an ordinary normal distribution, N(0,\sigma^2). Then, Y=, X, follows a half-normal distribution. Thus, the ha ...
, when \alpha = 2 and \nu = 1/2 ; **
Weibull distribution In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice R ...
, when \alpha > 0 and \nu = 1 ; **
Stretched Exponential distribution Stretching is a form of physical exercise in which a specific muscle or tendon (or muscle group) is deliberately flexed or stretched in order to improve the muscle's felt elasticity and achieve comfortable muscle tone. The result is a feeling ...
, when \alpha > 0 and \nu = 1/\alpha ;


See also

*
Giorgio Kaniadakis Kaniadakis Giorgio ( el, Κανιαδάκης Γεώργιος; born on 5 June 1957 in Chania-Crete, Greece) a Greek-Italian physicist, is a Full Professor of Theoretical Physics at Politecnico di Torino (Italy) and is credited with introducing ...
* Kaniadakis statistics *
Kaniadakis distribution In statistics, a Kaniadakis distribution (also known as κ-distribution) is a statistical distribution that emerges from the Kaniadakis statistics. There are several families of Kaniadakis distributions related to different constraints used in t ...
* 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