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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 $q\to 1$.

Characterization

Probability density function

The q-exponential distribution has the probability density function

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

where

$e_q(x)=[1+(1-q)x{]}^{\frac{1}{1-q}}$

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

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

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:

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

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:

$\text{If }X\sim \text{qExp}(q,\lambda )\text{ and }Y\sim [\text{Pareto}(x_m=\frac{1}{\lambda (q-1)},\alpha =\frac{2-q}{q-1})-x_m],\text{ then }X\sim Y$

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

$X=\frac{-q^{\prime }{\text{ ln}}_{q^{\prime }}(U)}{\lambda }\sim \text{qExp}(q,\lambda )$

where ${\text{ln}}_{q^{\prime }}$ is the q-logarithm and $q^{\prime }=\frac{1}{2-q}$

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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