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The q-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon Entropy. The exponential distribution is recovered as $q\rightarrow 1$.

Characterization

Probability density function

The q-exponential distribution has the probability density function

${(2-q)\lambda e_{q}^{-\lambda x}}$

where

$e_{q}(x)=[1+(1-q)x]^{1 \over 1-q}$

is the q-exponential.

Derivation

In a similar procedure to how the exponential distribution can be derived using the standard Boltzmann–Gibbs entropy or Shannon entropy and constraining the domain of the variable to be positive, the q-exponential distribution can be derived from a maximization of the Tsallis Entropy subject to the appropriate constraints.

Relationship to other distributions

The q-exponential is a special case of the Generalized Pareto distribution where

$\mu =0~,~\xi ={{q-1} \over {2-q}}~,~\sigma ={1 \over {\lambda (2-q)}}$

The q-exponential is the generalization of the Lomax distribution (Pareto Type II), as it extends this distribution to the cases of finite support. The Lomax parameters are:

$\alpha ={{2-q} \over {q-1}}~,~\lambda _{lomax}={1 \over {\lambda (q-1)}}$

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

${\text{If }}X\sim {\mbox{qExp}}(q,\lambda ){\text{ and }}Y\sim \left[{\text{Pareto}}\left(x_{m}={1 \over {\lambda (q-1)}},\alpha ={{2-q} \over {q-1}}\right)-x_{m}\right],{\text{ then }}X\sim Y\,$

Generating random deviates

Random deviates can be drawn using Inverse transform sampling. Given a variable U that is uniformly distributed on the interval (0,1), then

$X={{-q'{\text{ ln}}_{q'}(U)} \over \lambda }\sim {\mbox{qExp}}(q,\lambda )$

where ${\text{ln}}_{q'}$ is the q-logarithm and $q'={1 \over {2-q}}$

Applications

Economics (econophysics)

The q-exponential distribution has been used to describe the distribution of wealth (assets) between individuals.

See also

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

Notes

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Further reading

• Juniper, J. (2007) "The Tsallis Distribution and Generalised Entropy: Prospects for Future Research into Decision-Making under Uncertainty", Centre of Full Employment and Equity, The University of Newcastle, Australia

External links

• Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions

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