From Wikipedia, the free encyclopedia
Continuous probability 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.
Definitions
Probability density function
The Kaniadakis κ-Gamma distribution has the following
probability density function:
[1]
valid for , where is the entropic index associated with the
Kaniadakis entropy, , is the scale parameter, and is the shape parameter.
The ordinary
generalized Gamma distribution is recovered as : .
Cumulative distribution function
The
cumulative distribution function of κ-Gamma distribution assumes the form:
valid for , where . The cumulative
Generalized Gamma distribution is recovered in the classical limit .
Properties
Moments and mode
The κ-Gamma distribution has
moment of order given by
[1]
The moment of order of the κ-Gamma distribution is finite for .
The mode is given by:
Asymptotic behavior
The κ-Gamma distribution behaves
asymptotically as follows:
[1]
Related distributions
- The κ-Gamma distributions is a generalization of:
- A κ-Gamma distribution corresponds to several probability distributions when , such as:
-
Gamma distribution, when ;
-
Exponential distribution, when ;
-
Erlang distribution, when and positive integer;
-
Chi-Squared distribution, when and half integer;
-
Nakagami distribution, when and ;
-
Rayleigh distribution, when and ;
-
Chi distribution, when and half integer;
- Maxwell distribution, when and ;
-
Half-Normal distribution, when and ;
-
Weibull distribution, when and ;
-
Stretched Exponential distribution, when and ;
See also
References
External links
|
---|
Discrete univariate | with finite support | |
---|
with infinite support | |
---|
|
---|
Continuous univariate | supported on a bounded interval | |
---|
supported on a semi-infinite interval | |
---|
supported on the whole real line | |
---|
with support whose type varies | |
---|
|
---|
Mixed univariate | |
---|
Multivariate (joint) | |
---|
Directional | |
---|
Degenerate and
singular | |
---|
Families | |
---|
|