q-Weibull distribution
Probability density function ![]() | |||
Cumulative distribution function ![]() | |||
Parameters |
[math]\displaystyle{ q \lt 2 }[/math] shape (real) [math]\displaystyle{ \lambda \gt 0 }[/math] rate (real) [math]\displaystyle{ \kappa\gt 0\, }[/math] shape (real) | ||
---|---|---|---|
Support |
[math]\displaystyle{ x \in [0; +\infty)\! \text{ for }q \ge 1 }[/math] [math]\displaystyle{ x \in [0; {\lambda \over {(1-q)^{1/\kappa}}}) \text{ for } q\lt 1 }[/math] | ||
[math]\displaystyle{ \begin{cases} (2-q)\frac{\kappa}{\lambda}\left(\frac{x}{\lambda}\right)^{\kappa-1}e_{q}^{-(x/\lambda)^{\kappa}} & x\geq0\\ 0 & x\lt 0\end{cases} }[/math] | |||
CDF | [math]\displaystyle{ \begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x\lt 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:[1]
- [math]\displaystyle{ 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\lt 0, \end{cases} }[/math]
where q < 2, [math]\displaystyle{ \kappa }[/math] > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
- [math]\displaystyle{ 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 \gt 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 q-exponential[1][2][3]
Cumulative distribution function
The cumulative distribution function of a q-Weibull random variable is:
- [math]\displaystyle{ \begin{cases}1- e_{q'}^{-(x/\lambda')^\kappa} & x\geq0\\ 0 & x\lt 0\end{cases} }[/math]
where
- [math]\displaystyle{ \lambda' = {\lambda \over (2-q)^{1 \over \kappa}} }[/math]
- [math]\displaystyle{ q' = {1 \over (2-q)} }[/math]
Mean
The mean of the q-Weibull distribution is
- [math]\displaystyle{ \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\lt 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\lt q\lt 1+\frac{1+2\kappa}{1+\kappa}\\ \infty & 1+\frac{\kappa}{\kappa+1}\le q\lt 2 \end{cases} }[/math]
where [math]\displaystyle{ B() }[/math] is the Beta function and [math]\displaystyle{ \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]\displaystyle{ \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 distributions [math]\displaystyle{ (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]\displaystyle{ \kappa }[/math] parameter. The Lomax parameters are:
- [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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
References
- ↑ 1.0 1.1 Picoli, S. Jr.; Mendes, R. S.; Malacarne, L. C. (2003). "q-exponential, Weibull, and q-Weibull distributions: an empirical analysis". Physica A: Statistical Mechanics and Its Applications 324 (3): 678–688. doi:10.1016/S0378-4371(03)00071-2. Bibcode: 2003PhyA..324..678P.
- ↑ Naudts, Jan (2010). "The q-exponential family in statistical physics". Journal of Physics: Conference Series 201 (1): 012003. doi:10.1088/1742-6596/201/1/012003. Bibcode: 2010JPhCS.201a2003N.
- ↑ Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics". Milan Journal of Mathematics 76: 307–328. doi:10.1007/s00032-008-0087-y. http://www.santafe.edu/media/workingpapers/06-05-016.pdf. Retrieved 9 June 2014.
![]() | Original source: https://en.wikipedia.org/wiki/Q-Weibull distribution.
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