Q-Weibull Distribution

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:

:$f(x;q,\lambda,\kappa) = \begin (2-q)\frac\left(\frac\right)^ e_q(-(x/\lambda)^)& x\geq0 ,\\ 0 & x<0, \end$

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

:e_q(x) = \begin
\exp(x) & \textq=1, \\ pt +(1-q)x & \textq \ne 1 \text 1+(1-q)x >0, \\ pt0^ & \textq \ne 1\text1+(1-q)x \le 0, \\ pt\end

is the ''q''-exponential

Cumulative distribution function

The cumulative distribution function of a ''q''-Weibull random variable is:
:$\begin1- e_^ & x\geq0\\ 0 & x<0\end$
where
:$\lambda' =$
:$q' =$

Mean

The mean of the ''q''-Weibull distribution is

:
\mu(q,\kappa,\lambda) =
\begin
\lambda\,\left(2+\frac+\frac\right)(1-q)^\,B\left +\frac,2+\frac\right q<1 \\
\lambda\,\Gamma(1+\frac) & q=1\\
\lambda\,(2 - q) (q-1)^\,B\left +\frac, -\left(1+\frac+\frac\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-tailed distributions $(q \ge 1+\frac)$.

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:
:$\alpha = ~,~ \lambda_\text =$

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 X \sim \operatorname(q,\lambda,\kappa = 1) \text Y \sim \left operatorname
\left(
x_m = , \alpha =
\right) -x_m
\right
\text X \sim Y \,

See also

* Constantino Tsallis
* Tsallis statistics
* Tsallis entropy
* Tsallis distribution
* ''q''-Gaussian

References

{{DEFAULTSORT:Q-Weibull Distribution
Statistical mechanics
Continuous distributions
Probability distributions with non-finite variance