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.[1] The exponential distribution is recovered as

q-exponential distribution
Probability density function
Parameters shape (real)
rate (real)
Otherwise undefined
Mode 0
Ex. kurtosis

Originally proposed by the statisticians George Box and David Cox in 1964,[2] and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.


Probability density function

The q-exponential distribution has the probability density function


is the q-exponential if q ≠ 1. When q = 1, eq(x) is just exp(x).


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

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:

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, if


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

where is the q-logarithm and


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[3]. It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.[4]

See also


  1. Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
  2. Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 0192611.
  3. Keith Briggs and Christian Beck. "Modelling train delays with q-exponential functions". Physica A. 378: 498–504. doi:10.1016/j.physa.2006.11.084.
  4. C. Sun; N. A. Sinitsyn (2016). "Landau-Zener extension of the Tavis-Cummings model: Structure of the solution". Phys. Rev. A. 94 (3): 033808. arXiv:1606.08430. Bibcode:2016PhRvA..94c3808S. doi:10.1103/PhysRevA.94.033808.

Further reading

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