q-exponential distribution

Probability density function

Parameters|  q<2 shape (real)
λ>0 rate (real)
Support|  x∈[0,∞) for q≥1
x∈[0,1λ(1−q)) for q<1
PDF|  (2−q)λeq−λx
CDF|  1−eq′−λx/q′ where q′=12−q
Mean|  1λ(3−2q) for q<32
Otherwise undefined
Median|  −q′lnq′(1/2)λ where q′=12−q
Mode|  0
Variance|  q−2(2q−3)2(3q−4)λ2 for q<43
Skewness|  25−4q3q−4q−2 for q<54
Kurtosis|  6−4q3+17q2−20q+6(q−2)(4q−5)(5q−6) for q<65

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->1.

Originally proposed by the statisticians George Box and David Cox in 1964,[2] and known as the reverse Box–Cox transformation for q=1−λ, a particular case of power transform in statistics.

## Contents

* 1 Characterization
* 1.1 Probability density function
* 2 Derivation
* 3 Relationship to other distributions
* 4 Generating random deviates
* 5 Applications
* 6 See also
* 7 Notes
* 8 Further reading
* 9 External links

## Characterization

### Probability density function

The q-exponential distribution has the probability density function

(2−q)λeq(−λx)

where

eq(x)=[1+(1−q)x]1/(1−q)

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

## 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

μ=0,ξ=q−12−q,σ=1λ(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:

α=2−qq−1,λLomax=1λ(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, if

X∼q−Exp⁡(q,λ) and Y∼[Pareto⁡(xm=1λ(q−1),α=2−qq−1)−xm],

then X∼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′lnq′(U)λ∼q−Exp⁡(q,λ)

where lnq′ is the q-logarithm and q′=12−q.

## 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.[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

* Constantino Tsallis
* Tsallis statistics
* Tsallis entropy
* Tsallis distribution
* q-copula
* q-Gaussian

## Notes

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.
3. ↑ Keith Briggs and Christian Beck (2007). "Modelling train delays with q-exponential functions". Physica A 378 (2): 498–504. doi:10.1016/j.physa.2006.11.084. Bibcode: 2007PhyA..378..498B.
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. doi:10.1103/PhysRevA.94.033808. Bibcode: 2016PhRvA..94c3808S.

## 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

* v
* t
* e

Tsallis statistics

* Constantino Tsallis
* Tsallis entropy
* Tsallis statistics
* Tsallis distribution
* q-Gaussian distribution
* q-exponential distribution
* q-Weibull distribution

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