The Kaniadakis Weibull distribution (or κ-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution. It is one example of a Kaniadakis κ-distribution. The κ-Weibull distribution has been adopted successfully for describing a wide variety of complex systems in seismology, economy, epidemiology, among many others.
Definitions
Probability density function The Kaniadakis κ-Weibull distribution is exhibits power-law right tails, and it has the following probability density function:
f κ ( x ) = α β x α − 1 1 + κ 2 β 2 x 2 α exp κ ( − β x α ) {\displaystyle f_{_{\kappa }}(x)={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\exp _{\kappa }(-\beta x^{\alpha })}
valid for x ≥ 0 {\displaystyle x\geq 0} , where | κ | < 1 {\displaystyle |\kappa |<1} is the entropic index associated with the Kaniadakis entropy, β > 0 {\displaystyle \beta >0} is the scale parameter, and α > 0 {\displaystyle \alpha >0} is the shape parameter or Weibull modulus. The Weibull distribution is recovered as κ → 0. {\displaystyle \kappa \rightarrow 0.}
Cumulative distribution function The cumulative distribution function of κ-Weibull distribution is given by F κ ( x ) = 1 − exp κ ( − β x α ) {\displaystyle F_{\kappa }(x)=1-\exp _{\kappa }(-\beta x^{\alpha })} valid for x ≥ 0 {\displaystyle x\geq 0} . The cumulative Weibull distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
Survival distribution and hazard functions The survival distribution function of κ-Weibull distribution is given by
S κ ( x ) = exp κ ( − β x α ) {\displaystyle S_{\kappa }(x)=\exp _{\kappa }(-\beta x^{\alpha })}
valid for x ≥ 0 {\displaystyle x\geq 0} . The survival Weibull distribution is recovered in the classical limit κ → 0 {\displaystyle \kappa \rightarrow 0} .
The hazard function of the κ-Weibull distribution is obtained through the solution of the κ-rate equation: S κ ( x ) d x = − h κ S κ ( x ) {\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)} with S κ ( 0 ) = 1 {\displaystyle S_{\kappa }(0)=1} , where h κ {\displaystyle h_{\kappa }} is the hazard function:
h κ = α β x α − 1 1 + κ 2 β 2 x 2 α {\displaystyle h_{\kappa }={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}}
The cumulative κ-Weibull distribution is related to the κ-hazard function by the following expression:
S κ = e − H κ ( x ) {\displaystyle S_{\kappa }=e^{-H_{\kappa }(x)}}
where
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