View source for Q-Weibull distribution

{{DISPLAYTITLE:''q''-Weibull distribution}}
{{Probability distribution
|  name       =''q''-Weibull distribution
|  type       =density
  | pdf_image  = [[File:QWeibull pdf.svg|350px|Graph of the ''q''-Weibull pdf]]
  | cdf_image  = [[File:QWeibull cdf.svg|350px|Graph of the ''q''-Weibull cdf]]
  |parameters =<math>q < 2 </math> [[shape parameter|shape]] ([[Real number|real]]) <br /> <math> \lambda > 0 </math> [[rate parameter|rate]] ([[Real number|real]]) <br/><math>\kappa>0\,</math> [[shape parameter|shape]] (real)|
  support    =<math>x \in [0; +\infty)\! \text{ for }q \ge 1 </math> <br /> <math> x \in [0; {\lambda \over {(1-q)^{1/\kappa}}}) \text{ for } q<1 </math>|
  pdf        =<math>\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}</math>|
  cdf        =<math>\begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x<0\end{cases}</math>|
  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:<ref name="Picoli2008">{{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 }}</ref>

:<math>
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}</math>

where ''q'' < 2, <math>\kappa</math> > 0 are ''[[shape parameter]]s'' and λ > 0 is the ''[[scale parameter]]'' of the distribution and

:<math>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}
</math>

is the [[Tsallis statistics#q-exponential|''q''-exponential]]<ref name="Picoli2008"/><ref name="Naudts2010">{{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 }}</ref><ref name="Umarov2008">{{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}}</ref>

===Cumulative distribution function===
The [[cumulative distribution function]] of a ''q''-Weibull [[random variable]] is:
:<math>\begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x<0\end{cases}</math>
where
:<math>\lambda' = {\lambda \over (2-q)^{1 \over \kappa}} </math>
:<math>q' = {1 \over (2-q)} </math>

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

:<math>
\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<q<1+\frac{1+2\kappa}{1+\kappa}\\
\infty & 1+\frac{\kappa}{\kappa+1}\le q<2
\end{cases}</math>

where <math>B()</math> is the [[Beta function]] and <math>\Gamma()</math> 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 <math>\kappa=1</math>

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 distribution]]s <math>(q \ge 1+\frac{\kappa}{\kappa+1})</math>.

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 <math>\kappa</math> parameter. The Lomax parameters are:
:<math> \alpha = { {2-q} \over {q-1}} ~,~ \lambda_\text{Lomax} = {1 \over {\lambda (q-1)}} </math>

As the Lomax distribution is a shifted version of the [[Pareto distribution]], the ''q''-Weibull for <math>\kappa=1</math> 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:
:<math>
\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 \, 
</math>

== See also ==

* [[Constantino Tsallis]]
* [[Tsallis statistics]]
* [[Tsallis entropy]]
* [[Tsallis distribution]]
* [[q-Gaussian|''q''-Gaussian]]

== References ==
{{Reflist}}

{{Tsallis}}
{{ProbDistributions|continuous-variable}}

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