Q-exponential distribution

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
Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.

Characterization

Probability density function

The q-exponential distribution has the probability density function
where
is the q-exponential if. When, e_q is just exp.

Derivation

In a similar procedure to how the exponential distribution can be derived, 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
The q-exponential is the generalization of the Lomax distribution, as it extends this distribution to the cases of finite support. The Lomax parameters are:
As the Lomax distribution is a shifted version of the Pareto distribution, the q-exponential is a shifted reparameterized generalization of the Pareto. When, the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if
then

Generating random deviates

Random deviates can be drawn using inverse transform sampling. Given a variable U that is uniformly distributed on the interval, then
where is the q-logarithm and

Applications

Being a power transform, it is a usual technique in statistics for stabilizing the variance, making the data more normal distribution-like and improving the validity of measures of association such as the Pearson correlation between variables.
It has been found to be an accurate model for train delays.
It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.