Tsallis distribution: Difference between revisions

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The Tsallis distribution, also known as a q-Gaussian, is a probability distribution arising from the optimisation of the Tsallis entropy. While it can be justified by an interesting alternative form of entropy, statistically it is a simple reparametrization of the Student's t-distribution introduced by W. Gosset in 1908 to describe small sample statistics. In Gosset's original presentation the degrees of freedom parameter $\nu$ was constrained to be a positive integer related to the sample size, but it is readily observed that Gosset's density function is valid for all real values of $\nu$ . The reparametrization introduces the alternative parameter $q$ related to $\nu$ by

q={\frac {(\nu +3)}{(\nu +1)}}

with inverse

\nu ={\frac {(3-q)}{(q-1)}}

It is sometimes argued that the distribution is a generalization of the Student to negative and or non-integer degrees of freedom. However, almost the theory of the Student extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of $\nu <0$ . Furthermore, the case $\nu <0$ has to be covered by the two non-intersecting regions $q<1$ and $q>3$ , so the original Student parametrisation is in many respects cleaner. The formula for the T-density function is given in many standard texts and it is a simple matter to confirm the above formulae. The best starting point is Gosset's original work, discussed in the article on William_Sealy_Gosset.

Applications

Physics

Geography

Economics (econophysics)

The Tsallis distribution has been used to describe the distribution of wealth (assets) between individuals ^[1].

Notes

^ Adrian A. Dragulescu Applications of physics to economics and finance: Money, income, wealth, and the stock market arXiv:cond-mat/0307341v2

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 :<math>\nu = \frac{(3-q)}{(q-1)}</math>
-It is sometimes argued that the distribution is a generalization of the Student to negative and or non-integer degrees of freedom. However, almost the theory of the Student extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of <math>\nu<0</math>. Furthermore, the case <math>\nu<0</math> has to be covered by the two non-intersecting regions <math>q<1</math> and <math>q>3</math>, so the original Student parametrisation is in many respects cleaner.
+It is sometimes argued that the distribution is a generalization of the Student to negative and or non-integer degrees of freedom. However, almost the theory of the Student extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of <math>\nu<0</math>. Furthermore, the case <math>\nu<0</math> has to be covered by the two non-intersecting regions <math>q<1</math> and <math>q>3</math>, so the original Student parametrisation is in many respects cleaner. The formula for the T-density function is given in many standard texts and it is a simple matter to confirm the above formulae. The best starting point is Gosset's original work, discussed in the article on [[William_Sealy_Gosset]].
 == Applications ==