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:

$f(x;q,\lambda ,\kappa )=\{\begin{array}{cc}(2-q)\frac{\kappa }{\lambda }{\left(\frac{x}{\lambda }\right)}^{\kappa -1}{e}_q(-(x/\lambda {)}^{\kappa })& x\ge 0,\\ 0& x<0,\end{array}$

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

$e_q(x)=\{\begin{array}{cc}\exp (x)& \text{if }q=1,\\ [1+(1-q)x{]}^{1/(1-q)}& \text{if }q\ne 1\text{ and }1+(1-q)x>0,\\ 0^{1/(1-q)}& \text{if }q\ne 1\text{ and }1+(1-q)x\le 0,\end{array}$

is the q-exponential

Cumulative distribution function

The cumulative distribution function of a q-Weibull random variable is:

$\{\begin{array}{cc}1-e_{q^{\prime }}^{-(x/{\lambda }^{\prime }{)}^{\kappa }}& x\ge 0\\ 0& x<0\end{array}$

where

${\lambda }^{\prime }=\frac{\lambda }{(2-q{)}^{\frac{1}{\kappa }}}$

$q^{\prime }=\frac{1}{(2-q)}$

Mean

The mean of the q-Weibull distribution is

$\mu (q,\kappa ,\lambda )=\{\begin{array}{cc}\lambda (2+\frac{1}{1-q}+\frac{1}{\kappa })(1-q{)}^{-\frac{1}{\kappa }}B[1+\frac{1}{\kappa },2+\frac{1}{1-q}]& q<1\\ \lambda \mathrm{\Gamma }(1+\frac{1}{\kappa })& q=1\\ \lambda (2-q)(q-1{)}^{-\frac{1+\kappa }{\kappa }}B[1+\frac{1}{\kappa },-(1+\frac{1}{q-1}+\frac{1}{\kappa })]& 1<q<1+\frac{1+2\kappa }{1+\kappa }\\ \mathrm{\infty }& 1+\frac{\kappa }{\kappa +1}\le q<2\end{array}$

where $B()$ is the Beta function and $\mathrm{\Gamma }()$ 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:

$\alpha =\frac{2-q}{q-1},{\lambda }_{\text{Lomax}}=\frac{1}{\lambda (q-1)}$

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:

$\text{If }X\sim 𝑞\mathrm{-}\mathrm{W}\mathrm{e}\mathrm{i}\mathrm{b}\mathrm{u}\mathrm{l}\mathrm{l}(q,\lambda ,\kappa =1)\text{ and }Y\sim [\mathrm{Pareto}(x_m=\frac{1}{\lambda (q-1)},\alpha =\frac{2-q}{q-1})-x_m],\text{ then }X\sim Y$

See also

• Constantino Tsallis
• Tsallis statistics
• Tsallis entropy
• Tsallis distribution
• q-Gaussian