Q-Weibull distribution


In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution. It is one example of a Tsallis distribution.

Characterization

Probability density function

The probability density function of a q-Weibull random variable is:
where q < 2, > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
is the q-exponential

Cumulative distribution function

The cumulative distribution function of a q-Weibull random variable is:
where

Mean

The mean of the q-Weibull distribution is
where is the Beta function and is the Gamma function. The expression for the mean is a continuous function of q over the range of definition for which it is finite.

Relationship to other distributions

The q-Weibull is equivalent to the Weibull distribution when q = 1 and equivalent to the q-exponential when
The q-Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support and to include heavy-tailed distributions.
The q-Weibull is a generalization of the Lomax distribution, as it extends this distribution to the cases of finite support and adds the parameter. The Lomax parameters are:
As the Lomax distribution is a shifted version of the Pareto distribution, the q-Weibull for is a shifted reparameterized generalization of the Pareto. When q > 1, the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically: