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κ-Weibull distribution
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0}$rateshape (real) ${\displaystyle \beta >0}$rate (real)
Support	${\displaystyle x\in [0,+\mathrm{\infty })}$
PDF	${\displaystyle \frac{\alpha \beta x^{\alpha -1}}{\sqrt{1+{\kappa }^2{\beta }^2{x}^{2\alpha }}}{\exp }_{\kappa }(-\beta x^{\alpha })}$
CDF	${\displaystyle 1-{\exp }_{\kappa }(-\beta x^{\alpha })}$
Quantile	${\displaystyle {\beta }^{-1/\alpha }[{\ln }_{\kappa }(\frac{1}{1-F_{\kappa }}){]}^{1/\alpha }}$
Median	${\displaystyle {\beta }^{-1/\alpha }({\ln }_{\kappa }(2){)}^{1/\alpha }}$
Mode	${\displaystyle {\beta }^{-1/\alpha }(\frac{{\alpha }^2+2{\kappa }^2(\alpha -1)}{2{\kappa }^2({\alpha }^2-{\kappa }^2)}\sqrt{1+\frac{4{\kappa }^2({\alpha }^2-{\kappa }^2)(\alpha -1{)}^2}{[{\alpha }^2+2{\kappa }^2(\alpha -1){]}^2}}-1{)}^{1/2\alpha }}$
Method of moments	${\displaystyle \frac{(2\kappa \beta {)}^{-m/\alpha }}{1+\kappa \frac{m}{\alpha }}\frac{\mathrm{\Gamma }(\frac{1}{2\kappa }-\frac{m}{2\alpha })}{\mathrm{\Gamma }(\frac{1}{2\kappa }+\frac{m}{2\alpha })}\mathrm{\Gamma }(1+\frac{m}{\alpha })}$

TheKaniadakis Weibull distribution (orκ-Weibull distribution) is aprobability distribution arising as a generalization of theWeibull distribution.^[1][2] It is one example of aKaniadakisκ-distribution. The κ-Weibull distribution has been adopted successfully for describing a wide variety ofcomplex systems in seismology, economy, epidemiology, among many others.

The Kaniadakisκ-Weibull distribution is exhibits power-law right tails, and it has the followingprobability density function:^[3]

${\displaystyle f_{{}_{\kappa }}(x)=\frac{\alpha \beta x^{\alpha -1}}{\sqrt{1+{\kappa }^2{\beta }^2{x}^{2\alpha }}}{\exp }_{\kappa }(-\beta x^{\alpha })}$

valid for${\displaystyle x\ge 0}$, where is the entropic index associated with theKaniadakis entropy,${\displaystyle \beta >0}$ is the scale parameter, and${\displaystyle \alpha >0}$ is the shape parameter orWeibull modulus.

TheWeibull distribution is recovered as${\displaystyle \kappa \to 0.}$

Thecumulative distribution function ofκ-Weibull distribution is given by

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

valid for${\displaystyle x\ge 0}$. The cumulativeWeibull distribution is recovered in the classical limit${\displaystyle \kappa \to 0}$.

The survival distribution function ofκ-Weibull distribution is given by

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

valid for${\displaystyle x\ge 0}$. The survivalWeibull distribution is recovered in the classical limit${\displaystyle \kappa \to 0}$.

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

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

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

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

The cumulativeκ-Weibull distribution is related to theκ-hazard function by the following expression:

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

where

${\displaystyle H_{\kappa }(x)={\int }_0^x{h}_{\kappa }(z)dz}$

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

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

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

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

The median and the mode are:

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

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

Thequantiles are given by the following expression

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

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

TheGini coefficient is:^[3]

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

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

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

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

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

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

• Ineconomy, for analyzingpersonal 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]
• Inseismology, the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing theGutenberg–Richter law,^[6] and the interval distributions of seismic data, modeling extreme-event return intervals.^[7][8]
• Inepidemiology, 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
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