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.

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

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} $|undefined"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.