Q-Weibull distribution: Difference between revisions
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===Probability density function=== |
===Probability density function=== |
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The [[probability density function]] of a q-Weibull [[random variable]] is<ref name="Picoli2008">{{cite journal |last=Picoli |first=S. Jr. |last2=Mendes |first2=R. S. |last3=Malacarne |first3=L. C. |date=2008 |title=q-exponential, Weibull, and q-Weibull distributions: an empirical analysis |url=http://arxiv.org/pdf/cond-mat/0301552.pdf |journal=arXiv:cond-mat |publisher= |volume= |issue= |pages= |doi= |accessdate=9 June 2014}}</ref>: |
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The [[probability density function]] of a q-Weibull [[random variable]] is: |
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is the [[Tsallis statistics#q-exponential|q-exponential]]<ref name="Naudts2010">{{cite journal |last=Naudts |first=Jan |date=2010 |title=The q-exponential family in statistical physics |url=http://iopscience.iop.org/1742-6596/201/1/012003/pdf/1742-6596_201_1_012003.pdf |journal=J. Phys. Conf. Ser. |publisher=IOP Publishing |volume=201 |issue= |pages= |doi=10.1088/1742-6596/201/1/012003 |accessdate=9 June 2014}}</ref><ref name="Umarov2008">{{cite journal |last= |first= |last2= |first2= |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 j. math. |publisher= |volume=76 |issue= |pages= |doi=10.1007/s00032-008-0087-y |accessdate=9 June 2014}}</ref><ref name="Picoli2008"/> |
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is the [[Tsallis statistics#q-exponential|q-exponential]]. |
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===Cumulative distribution function=== |
===Cumulative distribution function=== |
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The [[cumulative distribution function]] of a q-Weibull [[random variable]] is: |
The [[cumulative distribution function]] of a q-Weibull [[random variable]] is: |
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== References == |
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<references/> |
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== See also == |
== See also == |
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Revision as of 00:35, 11 June 2014
Probability density function | |||
Cumulative distribution function | |||
Parameters |
shape (real) rate (real) shape (real) | ||
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Support |
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CDF | |||
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]:
where q < 2, > 0 are shape parameters and λ > 0 is the scale parameter of the distribution and
is the q-exponential[2][3][1]
Cumulative distribution function
The cumulative distribution function of a q-Weibull random variable is:
where
Mean
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 tail 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:
References
- ^ a b Picoli, S. Jr.; Mendes, R. S.; Malacarne, L. C. (2008). "q-exponential, Weibull, and q-Weibull distributions: an empirical analysis" (PDF). arXiv:cond-mat. Retrieved 9 June 2014.
- ^ Naudts, Jan (2010). "The q-exponential family in statistical physics" (PDF). J. Phys. Conf. Ser. 201. IOP Publishing. doi:10.1088/1742-6596/201/1/012003. Retrieved 9 June 2014.
- ^ "On a q-Central Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan j. math. 76. 2008. doi:10.1007/s00032-008-0087-y. Retrieved 9 June 2014.