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:
f κ ( x ) = ( 1 + κ ν ) ( 2 κ ) ν Γ ( 1 2 κ + ν 2 ) Γ ( 1 2 κ − ν 2 ) α β ν Γ ( ν ) x α ν − 1 exp κ ( − β x α ) {\displaystyle 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 ≥ 0 {\displaystyle x\geq 0} , where 0 ≤ | κ | < 1 {\displaystyle 0\leq |\kappa |<1} is the entropic index associated with the Kaniadakis entropy, 0 < ν < 1 / κ {\displaystyle 0<\nu <1/\kappa } , β > 0 {\displaystyle \beta >0} is the scale parameter, and α > 0 {\displaystyle \alpha >0} is the shape parameter. The ordinary generalized Gamma distribution is recovered as κ → 0 {\displaystyle \kappa \rightarrow 0} : f 0 ( x ) = | α | β ν Γ ( ν ) x α ν − 1 exp κ ( − β x α ) {\displaystyle 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:
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