q-exponential distribution

Probability density function

Parameters|  q < 2 {\displaystyle q<2} shape (real)
λ > 0 {\displaystyle \lambda >0} rate (real)
Support|  x ∈ [ 0 , ∞ ) for q ≥ 1 {\displaystyle x\in [0,\infty ){\text{ for }}q\geq 1}
x ∈ [ 0 , 1 λ ( 1 − q ) ) for q < 1 {\displaystyle x\in \left[0,{\frac {1}{\lambda (1-q)}}\right){\text{ for }}q<1}
PDF|  ( 2 − q ) λ e q ( − λ x ) {\displaystyle (2-q)\lambda e_{q}(-\lambda x)}
CDF|  1 − e q ′ − λ x / q ′ where q ′ = 1 2 − q {\displaystyle 1-e_{q'}^{-\lambda x/q'}{\text{ where }}q'={\frac {1}{2-q}}}
Mean|  1 λ ( 3 − 2 q ) for q < 3 2 {\displaystyle {\frac {1}{\lambda (3-2q)}}{\text{ for }}q<{\frac {3}{2}}}
Otherwise undefined
Median|  − q ′ ln q ′ ⁡ ( 1 / 2 ) λ where q ′ = 1 2 − q {\displaystyle {\frac {-q'\ln _{q'}(1/2)}{\lambda }}{\text{ where }}q'={\frac {1}{2-q}}}
Mode|  0
Variance|  q − 2 ( 2 q − 3 ) 2 ( 3 q − 4 ) λ 2 for q < 4 3 {\displaystyle {\frac {q-2}{(2q-3)^{2}(3q-4)\lambda ^{2}}}{\text{ for }}q<{\frac {4}{3}}}
Skewness|  2 5 − 4 q 3 q − 4 q − 2 for q < 5 4 {\displaystyle {\frac {2}{5-4q}}{\sqrt {\frac {3q-4}{q-2}}}{\text{ for }}q<{\frac {5}{4}}}
Excess kurtosis|  6 − 4 q 3 + 17 q 2 − 20 q + 6 ( q − 2 ) ( 4 q − 5 ) ( 5 q − 6 ) for q < 6 5 {\displaystyle 6{\frac {-4q^{3}+17q^{2}-20q+6}{(q-2)(4q-5)(5q-6)}}{\text{ for }}q<{\frac {6}{5}}}

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. {\displaystyle q\rightarrow 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 , {\displaystyle q=1-\lambda ,} a particular case of power transform in statistics.

## Characterization

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### Probability density function

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The q-exponential distribution has the probability density function

( 2 − q ) λ e q ( − λ x ) {\displaystyle (2-q)\lambda e_{q}(-\lambda x)}

where

e q ( x ) = [ 1 + ( 1 − q ) x ] 1 / ( 1 − q ) {\displaystyle e_{q}(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

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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

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

μ = 0 , ξ = q − 1 2 − q , σ = 1 λ ( 2 − q ) . {\displaystyle \mu =0,\quad \xi ={\frac {q-1}{2-q}},\quad \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:

α = 2 − q q − 1 , λ L o m a x = 1 λ ( q − 1 ) . {\displaystyle \alpha ={\frac {2-q}{q-1}},\quad \lambda _{\mathrm {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, if

X ∼ q - E x p ⁡ ( q , λ ) and Y ∼ [ Pareto ⁡ ( x m = 1 λ ( q − 1 ) , α = 2 − q q − 1 ) − x m ] , {\displaystyle X\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda ){\text{ and }}Y\sim \left[\operatorname {Pareto} \left(x_{m}={\frac {1}{\lambda (q-1)}},\alpha ={\frac {2-q}{q-1}}\right)-x_{m}\right],}

then  X ∼ Y . {\displaystyle X\sim Y.}

## Generating random deviates

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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 ′ ln q ′ ⁡ ( U ) λ ∼ q - E x p ⁡ ( q , λ ) {\displaystyle X={\frac {-q'\ln _{q'}(U)}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}

where  ln q ′ {\displaystyle \ln _{q'}} is the q-logarithm and  q ′ = 1 2 − q . {\displaystyle q'={\frac {1}{2-q}}.}

## Applications

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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

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

## Notes

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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 (2007). "Modelling train delays with q-exponential functions". Physica A. 378 (2): 498–504\. arXiv:physics/0611097. Bibcode:2007PhyA..378..498B. doi:10.1016/j.physa.2006.11.084. S2CID 107475.
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. S2CID 119317114.

## Further reading

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* 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

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

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Tsallis statistics

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Probability distributions (list)

Discrete
univariate|

| with finite
support|

* Benford
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|

with infinite
support|

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Continuous
univariate|

| supported on a
bounded interval|

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|

supported on a
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* Wilks's lambda

supported
on the whole
real line|

* Cauchy
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with support
whose type varies|

* Generalized chi-squared
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* Kaniadakis κ-exponential
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* Kaniadakis κ-Logistic
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* q-exponential
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* Tukey lambda

Mixed
univariate|

| continuous-
discrete|

* Rectified Gaussian

|

Multivariate
(joint)|

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Directional|

Univariate (circular) directional
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Bivariate (spherical)
Kent
Bivariate (toroidal)
Bivariate von Mises
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von Mises–Fisher
Bingham

Degenerate
and singular|

Degenerate
Dirac delta function
Singular
Cantor

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