QGaussian distribution
Probability density function


Parameters  shape (real) (real) 

Support  for for 
Mean  , otherwise undefined 
Median  
Mode  
Variance  
Skewness  
Ex. kurtosis 
The qGaussian is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints. It is one example of a Tsallis distribution. The qGaussian is a generalization of the Gaussian in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy.^{1} The normal distribution is recovered as .
The qGaussian has been applied to problems in the fields of statistical mechanics, geology, anatomy, astronomy, economics, finance, and machine learning. The distribution is often favored for its heavy tails in comparison to the Gaussian for . There is generalized qanalog of the classical central limit theorem^{2} in which the independence constraint for the i.i.d. variables is relaxed to an extent defined by the q parameter, with independence being recovered as q → 1. In analogy to the classical central limit theorem, an average of such random variables with fixed mean and variance tend towards the qGaussian distribution.
In the heavy tail regions, the distribution is equivalent to the Student's tdistribution with a direct mapping between q and the degrees of freedom. A practitioner using one of these distributions can therefore parameterize the same distribution in two different ways. The choice of the qGaussian form may arise if the system is nonextensive, or if there is lack of a connection to small samples sizes.
Contents
Characterization
Probability density function
The qGaussian has the probability density function ^{2}
where
is the qexponential and the normalization factor is given by
Entropy
Just as the normal distribution is the maximum information entropy distribution for fixed values of the first moment and second moment (with the fixed zeroth moment corresponding to the normalization condition), the qGaussian distribution is the maximum Tsallis entropy distribution for fixed values of these three moments.
Related distributions
Student's tdistribution
While it can be justified by an interesting alternative form of entropy, statistically it is a scaled reparametrization of the Student's tdistribution introduced by W. Gosset in 1908 to describe smallsample statistics. In Gosset's original presentation the degrees of freedom parameter 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 .^{citation needed} The scaled reparametrization introduces the alternative parameters which are related to .
Given a Student's t distribution with degrees of freedom, the equivalent qGaussian has
with inverse
Whenever , the function is simply a scaled version of Student's t distribution.
It is sometimes argued that the distribution is a generalization of Student's t distribution to negative and or noninteger degrees of freedom. However, the theory of Student's t distribution extends trivially to all real degrees of freedom, where the support of the distribution is now compact rather than infinite in the case of .^{citation needed}
Threeparameter version
As with many distributions centered on zero, the qgaussian can be trivially extended to include a location parameter . The density then becomes defined by
Generating random deviates
The Box–Muller transform has been generalized to allow random sampling from qgaussians.^{3} The standard Box–Muller technique generates pairs of independent normally distributed variables from equations of the following form.
The generalized Box–Muller technique can generates pairs of qgaussian deviates that are not independent. In practice, only a single deviate will be generated from a pair of uniformly distributed variables. The following formula will generate deviates from a qGaussian with specified parameter q and
Where is the qlogarithm and
These deviates can be transformed to generate deviates from an arbitrary qGaussian by
Applications
Physics
It has been shown that the momentum distribution of cold atoms in dissipative optical lattices is a qGaussian^{4}
Finance
Financial return distributions in the New York Stock Exchange, NASDAQ and elsewhere are often interpreted as qGaussians.^{5}^{6}
See also
 Constantino Tsallis
 Tsallis statistics
 Tsallis entropy
 Tsallis distribution
 qexponential distribution
Notes
 ^ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanicsan overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
 ^ ^{a} ^{b} Umarov, Sabir; Tsallis, Constantino; Steinberg, Stanly (2008). "On a qCentral Limit Theorem Consistent with Nonextensive Statistical Mechanics" (PDF). Milan j. math. (Birkhauser Verlag) 76: 307–328. doi:10.1007/s000320080087y. Retrieved 20110727.
 ^ W. Thistleton, J.A. Marsh, K. Nelson and C. Tsallis, Generalized Box–Muller method for generating qGaussian random deviates, IEEE Transactions on Information Theory 53, 4805 (2007)
 ^ Douglas, P.; Bergamini, S.; Renzoni, F. (2006). "Tunable Tsallis Distributions in Dissipative Optical Lattices". Physical Review Letters 96 (11). doi:10.1103/PhysRevLett.96.110601.
 ^ L.Borland, Option pricing formulas based on a nonGaussian stock price model, Phys. Rev. Lett. 89, 098701 (2002)
 ^ L. Borland, The pricing of stock options, in Nonextensive Entropy Interdisciplinary Applications, eds. M. GellMann and C. Tsallis (Oxford University Press, New York, 2004)
Further reading
 Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into DecisionMaking under Uncertainty", Centre of Full Employment and Equity, The University of Newcastle, Australia
External links
