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.

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	$\beta ^{-1/\alpha }{\Bigg [}\ln _{\kappa }{\Bigg (}{\frac {1}{1-F_{\kappa }}}{\Bigg )}{\Bigg ]}^{1/\alpha }$
Median	$\beta ^{-1/\alpha }{\Bigg (}\ln _{\kappa }(2){\Bigg )}^{1/\alpha }$
Mode	$\beta ^{-1/\alpha }{\Bigg (}{\frac {\alpha ^{2}+2\kappa ^{2}(\alpha -1)}{2\kappa ^{2}(\alpha ^{2}-\kappa ^{2})}}{\sqrt {1+{\frac {4\kappa ^{2}(\alpha ^{2}-\kappa ^{2})(\alpha -1)^{2}}{[\alpha ^{2}+2\kappa ^{2}(\alpha -1)]^{2}}}}}-1{\Bigg )}^{1/2\alpha }$
Method of moments	${\frac {(2\kappa \beta )^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big (}1+{\frac {m}{\alpha }}{\Big )}$

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

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

The median and the mode are:

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

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

Quantiles

The quantiles are given by the following expression

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

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.

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