The Kaniadakis Weibull distribution (or ú-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution. It is one example of a Kaniadakis ú-distribution. The ú-Weibull distribution has been adopted successfully for describing a wide variety of complex systems in seismology, economy, epidemiology, among many others.
The Kaniadakis ú-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
The cumulative distribution function of ú-Weibull distribution is given by<blockquote></blockquote>valid for . The cumulative Weibull distribution is recovered in the classical limit .
The survival distribution function of ú-Weibull distribution is given by
valid for . The survival Weibull distribution is recovered in the classical limit .
The hazard function of the ú-Weibull distribution is obtained through the solution of the ú-rate equation:<blockquote></blockquote>with , where is the hazard function:
The cumulative ú-Weibull distribution is related to the ú-hazard function by the following expression:
where
is the cumulative ú-hazard function. The cumulative hazard function of the Weibull distribution is recovered in the classical limit : .
The ú-Weibull distribution has moment of order given by
The median and the mode are:
The quantiles are given by the following expression<blockquote></blockquote>with .
The Gini coefficient is:<blockquote></blockquote>
The ú-Weibull distribution II behaves asymptotically as follows:
The ú-Weibull distribution has been applied in several areas, such as: