Kaniadakis Weibull distribution

Continuous probability distribution

The topic of this article may not meet Misplaced Pages's general notability guideline. Please help to demonstrate the notability of the topic by citing reliable secondary sources that are independent of the topic and provide significant coverage of it beyond a mere trivial mention. If notability cannot be shown, the article is likely to be merged, redirected, or deleted.
Find sources: "Kaniadakis Weibull distribution" – news · newspapers · books · scholar · JSTOR (February 2023) (Learn how and when to remove this message)

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

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

valid for $x\geq 0$ , where $|\kappa |<1$ is the entropic index associated with the Kaniadakis entropy, $\beta >0$ is the scale parameter, and $\alpha >0$ is the shape parameter or Weibull modulus.

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

Cumulative distribution function

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

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

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

Survival distribution and hazard functions

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

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

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

Comparison between the Kaniadakis κ-Weibull probability function and its cumulative.

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

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

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

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_{\kappa }=e^{-H_{\kappa }(x)}

where

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

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 $\kappa \rightarrow 0$ : $H(x)=\beta x^{\alpha }$ .

Properties

Moments, median and mode

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

\operatorname {E} ={\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:

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

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}}{^{2}}}}}-1{\Bigg )}^{1/2\alpha }\quad (\alpha >1)

Quantiles

The quantiles are given by the following expression

$x_{\textrm {quantile}}(F_{\kappa })=\beta ^{-1/\alpha }{\Bigg }^{1/\alpha }$

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

Gini coefficient

The Gini coefficient is:

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

Asymptotic behavior

The κ-Weibull distribution II behaves asymptotically as follows:

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

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

Related distributions

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

Applications

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.
In seismology, the κ-Weibull represents the statistical distribution of magnitude of the earthquakes distributed across the Earth, generalizing the Gutenberg–Richter law, and the interval distributions of seismic data, modeling extreme-event return intervals.
In epidemiology, the κ-Weibull distribution presents a universal feature for epidemiological analysis.