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:

 

valid for , where  is the entropic index associated with the Kaniadakis entropy,  is the scale parameter, and  is the shape parameter or Weibull modulus. The Weibull distribution is recovered as

Cumulative distribution function 
The cumulative distribution function of κ-Weibull distribution is given byvalid for . The cumulative Weibull distribution is recovered in the classical limit . Survival distribution and hazard functions 
The survival distribution function of κ-Weibull distribution is given by

valid for . The survival Weibull distribution is recovered in the classical limit . The hazard function of the κ-Weibull distribution is obtained through the solution of the κ-rate equation:with  , where   is the hazard function:

The cumulative κ-Weibull distribution is related to the κ-hazard function by the following expression:

where

is the cumulative κ-hazard function. The cumulative hazard function of the Weibull distribution is recovered in the classical limit :  . Properties

Moments, median and mode 
The κ-Weibull distribution has moment of order  given by

The median and the mode are:

Quantiles 
The quantiles are given by the following expressionwith . Gini coefficient 
The Gini coefficient is:

Asymptotic behavior 
The κ-Weibull distribution II behaves asymptotically as follows:

Related distributions
The κ-Weibull distribution is a generalization of:
κ-Exponential distribution of type II, when ;
Exponential distribution when  and . A κ-Weibull distribution corresponds to a κ-deformed Rayleigh distribution when  and a Rayleigh distribution when  and . 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

References

External links
Kaniadakis Statistics on arXiv.org

Statistics
Probability distributions
Continuous distributions
Exponential family distributions
Survival analysis