Q-Weibull distribution

{{Probability distribution

| name =q-Weibull distribution

| type =density

|parameters = $q < 2$ shape (real)
$\lambda > 0$ rate (real)
$\kappa>0\,$ shape (real)|

support = $x \in [0; +\infty)\! \text{ for }q \ge 1$
$x \in [0; {\lambda \over {(1-q)^{1/\kappa}}}) \text{ for } q<1$ |

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

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

mean =(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:{{cite journal |last1=Picoli |first1=S. Jr. |last2=Mendes |first2=R. S. |last3=Malacarne |first3=L. C. |date=2003 |title=q-exponential, Weibull, and q-Weibull distributions: an empirical analysis |arxiv=cond-mat/0301552 |journal= Physica A: Statistical Mechanics and Its Applications|volume= 324|issue= 3|pages= 678–688|doi= 10.1016/S0378-4371(03)00071-2|bibcode=2003PhyA..324..678P|s2cid=119361445 }}

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

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{{cite journal |last=Naudts |first=Jan |date=2010 |title=The q-exponential family in statistical physics |journal= Journal of Physics: Conference Series|volume=201 |issue=1 |pages= 012003|doi=10.1088/1742-6596/201/1/012003 |arxiv=0911.5392 |bibcode=2010JPhCS.201a2003N |s2cid=119276469 }}{{cite journal |last2= Tsallis|first2= Constantino|last3= Steinberg|first3= Stanly|date=2008 |title=On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics |url=http://www.santafe.edu/media/workingpapers/06-05-016.pdf|journal= Milan Journal of Mathematics|volume=76 |pages= 307–328|doi=10.1007/s00032-008-0087-y |access-date=9 June 2014|last1= Umarov|first1= Sabir|s2cid= 55967725}}

=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-tailed 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:

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

References

Category:Statistical mechanics

Category:Continuous distributions

Category:Probability distributions with non-finite variance