# q-Weibull distribution

Parameters Probability density function Cumulative distribution function $q < 2$ shape (real) $\lambda > 0$ rate (real) $\kappa>0\,$ shape (real) $x \in [0; +\infty)\! \text{ for }q \ge 1$ $x \in [0; {\lambda \over {(1-q)^{1/\kappa}}}) \text{ for } q<1$ $\begin{cases} (2-q)\frac{\kappa}{\lambda}\left(\frac{x}{\lambda}\right)^{\kappa-1}e_{q}^{-(x/\lambda)^{\kappa}} & x\geq0\\ 0 & x<0\end{cases}$ $\begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x<0\end{cases}$ (see article)

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

## Characterization

### Probability density function

The probability density function of a q-Weibull random variable is:[1]

$f(x;q,\lambda,\kappa) = \begin{cases} (2-q)\frac{\kappa}{\lambda}\left(\frac{x}{\lambda}\right)^{\kappa-1}e_{q}^{-(x/\lambda)^{\kappa}} & x\geq0 ,\\ 0 & x<0, \end{cases}$

where q < 2, $\kappa$ > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and

$e_q(x) = \begin{cases} \exp(x) & \text{if }q=1, \\[6pt] [1+(1-q)x]^{1/(1-q)} & \text{if }q \ne 1 \text{ and } 1+(1-q)x >0, \\[6pt] 0^{1/(1-q)} & \text{if }q \ne 1\text{ and }1+(1-q)x \le 0, \\[6pt] \end{cases}$

is the q-exponential[1][2][3]

### Cumulative distribution function

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

$\begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x<0\end{cases}$

where

$\lambda' = {\lambda \over (2-q)^{1 \over \kappa}}$
$q' = {1 \over (2-q)}$

## Mean

The mean of the q-Weibull distribution is

$\mu(q,\kappa,\lambda) = \begin{cases} \lambda\,\left(2+\frac{1}{1-q}+\frac{1}{\kappa}\right)(1-q)^{-\frac{1}{\kappa}}\,B\left[1+\frac{1}{\kappa},2+\frac{1}{1-q}\right]& q<1 \\ \lambda\,\Gamma(1+\frac{1}{\kappa}) & q=1\\ \lambda\,(2 - q) (q-1)^{-\frac{1+\kappa}{\kappa}}\,B\left[1+\frac{1}{\kappa}, -\left(1+\frac{1}{q-1}+\frac{1}{\kappa}\right)\right] & 1

where $B()$ is the Beta function and $\Gamma()$ 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 $\kappa=1$

The q-Weibull is a generalization of the Weibull, as it extends this distribution to the cases of finite support (q < 1) and to include heavy tail distributions $(q \ge 1+\frac{\kappa}{\kappa+1})$.

The q-Weibull is a generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support and adds the $\kappa$ parameter. The Lomax parameters are:

$\alpha = { {2-q} \over {q-1}} ~,~ \lambda_\text{Lomax} = {1 \over {\lambda (q-1)}}$

As the Lomax distribution is a shifted version of the Pareto distribution, the q-Weibull for $\kappa=1$ 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:

$\text{If } X \sim \mathrm{qWeibull}(q,\lambda,\kappa = 1) \text{ and } Y \sim \left[\text{Pareto} \left( x_m = {1 \over {\lambda (q-1)}}, \alpha = { {2-q} \over {q-1}} \right) -x_m \right], \text{ then } X \sim Y \,$