Kaniadakis Weibull distribution

The Kaniadakis Weibull distribution (or κ-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution.^{[1] [2]} 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.

The Kaniadakis κ-Weibull distribution is exhibits power-law right tails, and it has the following probability density function:^[3]

${\displaystyle f_{_{\kappa }}(x)={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}\exp _{\kappa }(-\beta x^{\alpha })}${\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 ${\displaystyle x\geq 0}${\displaystyle x\geq 0}, where ${\displaystyle |\kappa |<1}${\displaystyle |\kappa |<1} is the entropic index associated with the Kaniadakis entropy, ${\displaystyle \beta >0}${\displaystyle \beta >0} is the scale parameter, and ${\displaystyle \alpha >0}${\displaystyle \alpha >0} is the shape parameter or Weibull modulus.

The Weibull distribution is recovered as ${\displaystyle \kappa \rightarrow 0.}${\displaystyle \kappa \rightarrow 0.}

The cumulative distribution function of κ-Weibull distribution is given by

${\displaystyle F_{\kappa }(x)=1-\exp _{\kappa }(-\beta x^{\alpha })}${\displaystyle F_{\kappa }(x)=1-\exp _{\kappa }(-\beta x^{\alpha })}

valid for ${\displaystyle x\geq 0}${\displaystyle x\geq 0}. The cumulative Weibull distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}.

The survival distribution function of κ-Weibull distribution is given by

${\displaystyle S_{\kappa }(x)=\exp _{\kappa }(-\beta x^{\alpha })}${\displaystyle S_{\kappa }(x)=\exp _{\kappa }(-\beta x^{\alpha })}

valid for ${\displaystyle x\geq 0}${\displaystyle x\geq 0}. The survival Weibull distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}.

The hazard function of the κ-Weibull distribution is obtained through the solution of the κ-rate equation:

${\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)}${\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)}

with ${\displaystyle S_{\kappa }(0)=1}${\displaystyle S_{\kappa }(0)=1}, where ${\displaystyle h_{\kappa }}${\displaystyle h_{\kappa }} is the hazard function:

${\displaystyle h_{\kappa }={\frac {\alpha \beta x^{\alpha -1}}{\sqrt {1+\kappa ^{2}\beta ^{2}x^{2\alpha }}}}}${\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:

${\displaystyle S_{\kappa }=e^{-H_{\kappa }(x)}}${\displaystyle S_{\kappa }=e^{-H_{\kappa }(x)}}

where

${\displaystyle H_{\kappa }(x)=\int _{0}^{x}h_{\kappa }(z)dz}${\displaystyle H_{\kappa }(x)=\int _{0}^{x}h_{\kappa }(z)dz}

${\displaystyle H_{\kappa }(x)={\frac {1}{\kappa }}{\textrm {arcsinh}}\left(\kappa \beta x^{\alpha }\right)}${\displaystyle H_{\kappa }(x)={\frac {1}{\kappa }}{\textrm {arcsinh}}\left(\kappa \beta x^{\alpha }\right)}

is the cumulative κ-hazard function. The cumulative hazard function of the Weibull distribution is recovered in the classical limit ${\displaystyle \kappa \rightarrow 0}${\displaystyle \kappa \rightarrow 0}: ${\displaystyle H(x)=\beta x^{\alpha }}${\displaystyle H(x)=\beta x^{\alpha }} .

The κ-Weibull distribution has moment of order ${\displaystyle m\in \mathbb {N} }${\displaystyle m\in \mathbb {N} } given by

${\displaystyle \operatorname {E} [X^{m}]={\frac {|2\kappa \beta |^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big (}1+{\frac {m}{\alpha }}{\Big )}}${\displaystyle \operatorname {E} [X^{m}]={\frac {|2\kappa \beta |^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big (}1+{\frac {m}{\alpha }}{\Big )}}

The median and the mode are:

${\displaystyle x_{\textrm {median}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg (}\ln _{\kappa }(2){\Bigg )}^{1/\alpha }}${\displaystyle x_{\textrm {median}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg (}\ln _{\kappa }(2){\Bigg )}^{1/\alpha }}

${\displaystyle x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\Bigg )}^{1/2\alpha }{\Bigg (}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }\quad (\alpha >1)}${\displaystyle x_{\textrm {mode}}=\beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\Bigg )}^{1/2\alpha }{\Bigg (}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }\quad (\alpha >1)}

The quantiles are given by the following expression

${\displaystyle x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }}${\displaystyle x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }}

with ${\displaystyle 0\leq F_{\kappa }\leq 1}${\displaystyle 0\leq F_{\kappa }\leq 1}.

The Gini coefficient is:^[3]

${\displaystyle \operatorname {G} _{\kappa }=1-{\frac {\alpha +\kappa }{\alpha +{\frac {1}{2}}\kappa }}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}-{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {1}{2\alpha }}{\Big )}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{2\alpha }}{\Big )}}}}${\displaystyle \operatorname {G} _{\kappa }=1-{\frac {\alpha +\kappa }{\alpha +{\frac {1}{2}}\kappa }}{\frac {\Gamma {\Big (}{\frac {1}{\kappa }}-{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{\kappa }}+{\frac {1}{2\alpha }}{\Big )}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {1}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {1}{2\alpha }}{\Big )}}}}

The κ-Weibull distribution II behaves asymptotically as follows:^[3]

${\displaystyle \lim _{x\to +\infty }f_{\kappa }(x)\sim {\frac {\alpha }{\kappa }}(2\kappa \beta )^{-1/\kappa }x^{-1-\alpha /\kappa }}${\displaystyle \lim _{x\to +\infty }f_{\kappa }(x)\sim {\frac {\alpha }{\kappa }}(2\kappa \beta )^{-1/\kappa }x^{-1-\alpha /\kappa }}

${\displaystyle \lim _{x\to 0^{+}}f_{\kappa }(x)=\alpha \beta x^{\alpha -1}}${\displaystyle \lim _{x\to 0^{+}}f_{\kappa }(x)=\alpha \beta x^{\alpha -1}}

• The κ-Weibull distribution is a generalization of:
  • κ-Exponential distribution of type II, when ${\displaystyle \alpha =1}${\displaystyle \alpha =1};
  • Exponential distribution when ${\displaystyle \kappa =0}${\displaystyle \kappa =0} and ${\displaystyle \alpha =1}${\displaystyle \alpha =1}.
• A κ-Weibull distribution corresponds to a κ-deformed Rayleigh distribution when ${\displaystyle \alpha =2}${\displaystyle \alpha =2} and a Rayleigh distribution when ${\displaystyle \kappa =0}${\displaystyle \kappa =0} and ${\displaystyle \alpha =2}${\displaystyle \alpha =2}.

The κ-Weibull distribution has been applied in several areas, such as:

• In economy, for analyzing personal income models, in order to accurately describing simultaneously the income distribution among the richest part and the great majority of the population.^{[1] [4] [5]}
• In seismology, the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing the Gutenberg–Richter law,^[6] and the interval distributions of seismic data, modeling extreme-event return intervals.^{[7] [8]}
• In epidemiology, the κ-Weibull distribution presents a universal feature for epidemiological analysis.^[9]

• Giorgio Kaniadakis
• Kaniadakis statistics
• Kaniadakis distribution
• Kaniadakis κ-Exponential distribution
• Kaniadakis κ-Gaussian distribution
• Kaniadakis κ-Gamma distribution
• Kaniadakis κ-Logistic distribution
• Kaniadakis κ-Erlang distribution

• Kaniadakis Statistics on arXiv.org

• Continuous distributions
• Exponential family distributions
• Survival analysis

• Articles with short description
• Short description matches Wikidata
• Articles with topics of unclear notability from February 2023
• All articles with topics of unclear notability