q-Weibull distribution

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q-Weibull distribution
Probability density function
Graph of the q-Weibull pdf
Cumulative distribution function
Graph of the q-Weibull cdf
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[edit]

Probability density function[edit]

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[edit]

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[edit]

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<q<1+\frac{1+2\kappa}{1+\kappa}\\
\infty & 1+\frac{\kappa}{\kappa+1}\le q<2
\end{cases}

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[edit]

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 \,

See also[edit]

References[edit]

  1. ^ a b Picoli, S. Jr.; Mendes, R. S.; Malacarne, L. C. (2008). "q-exponential, Weibull, and q-Weibull distributions: an empirical analysis". arXiv:cond-mat. Retrieved 9 June 2014. 
  2. ^ Naudts, Jan (2010). "The q-exponential family in statistical physics". J. Phys. Conf. Ser. (IOP Publishing) 201. doi:10.1088/1742-6596/201/1/012003. Retrieved 9 June 2014. 
  3. ^ "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics". Milan j. math. 76. 2008. doi:10.1007/s00032-008-0087-y. Retrieved 9 June 2014. 

Notes[edit]