The Kaniadakis Weibull distribution (or κ-Weibull distribution) is a probability distribution arising as a generalization of the Weibull distribution.

[1][2] 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.

The Kaniadakis κ-Weibull distribution is exhibits power-law right tails, and it has the following probability density function:[3] valid for

is the entropic index associated with the Kaniadakis entropy,

β > 0

is the scale parameter, and

is the shape parameter or Weibull modulus.

The Weibull distribution is recovered as

The cumulative distribution function of κ-Weibull distribution is given by

exp

⁡ ( − β

valid for

The cumulative Weibull distribution is recovered in the classical limit

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:

{\displaystyle {\frac {S_{\kappa }(x)}{dx}}=-h_{\kappa }S_{\kappa }(x)}

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

( x ) = β

The κ-Weibull distribution has moment of order

given by The median and the mode are:

The quantiles are given by the following expression

β

ln

The Gini coefficient is:[3]

α + κ

The κ-Weibull distribution II behaves asymptotically as follows:[3] The κ-Weibull distribution has been applied in several areas, such as: