# 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 $x\geq 0$, 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 $\kappa \rightarrow 0$.
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
    $\operatorname {E} [X^{m}]={\frac {|2\kappa \beta |^{-m/\alpha }}{1+\kappa {\frac {m}{\alpha }}}}{\frac {\Gamma {\Big{\frac {1}{2\kappa }}-{\frac {m}{2\alpha }}{\Big )}}{\Gamma {\Big{\frac {1}{2\kappa }}+{\frac {m}{2\alpha }}{\Big )}}}\Gamma {\Big1+{\frac {m}{\alpha }}{\Big )}$
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.