Kaniadakis Gamma Distribution
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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 ...
.


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
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
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
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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