q-Weibull distribution
Probability density function
Graph of the q-Weibull pdf
Cumulative distribution function
Graph of the q-Weibull cdf
Parameters shape (real)
rate (real)
shape (real)
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.


Probability density function

The probability density function of a q-Weibull random variable is:[1]

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

is the q-exponential[1][2][3]

Cumulative distribution function

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



The mean of the q-Weibull distribution is

where is the Beta function and 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

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 .

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 parameter. The Lomax parameters are:

As the Lomax distribution is a shifted version of the Pareto distribution, the q-Weibull for 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:

See also


  1. ^ a b 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. arXiv:cond-mat/0301552. Bibcode:2003PhyA..324..678P. doi:10.1016/S0378-4371(03)00071-2. S2CID 119361445.
  2. ^ Naudts, Jan (2010). "The q-exponential family in statistical physics". Journal of Physics: Conference Series. 201 (1): 012003. arXiv:0911.5392. Bibcode:2010JPhCS.201a2003N. doi:10.1088/1742-6596/201/1/012003. S2CID 119276469.
  3. ^ Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan Journal of Mathematics. 76: 307–328. doi:10.1007/s00032-008-0087-y. S2CID 55967725. Retrieved 9 June 2014.