Kaniadakis Weibull distribution

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 $x\ge 0$, 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 $\kappa \to 0.$

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 $x\ge 0$. The cumulative Weibull distribution is recovered in the classical limit $\kappa \to 0$.

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 $x\ge 0$. The survival Weibull distribution is recovered in the classical limit $\kappa \to 0$.

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

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

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

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

Properties

Moments, median and mode

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

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

The median and the mode are:

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

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

Quantiles

The quantiles are given by the following expression

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

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

Gini coefficient

The Gini coefficient is:

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

Asymptotic behavior

The κ-Weibull distribution II behaves asymptotically as follows:

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

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

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.

See also

• 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

External links

• Kaniadakis Statistics on arXiv.org