q-exponential distribution

{{Probability distribution |

name =q-exponential distribution|

type =density|

pdf_image =File:The Probability Density Function of q-exponential distribution.svg|

parameters = $q < 2$ shape (real)
$\lambda > 0$ rate (real) |

support = $x \in [0, \infty) \text{ for } q \ge 1$
$x \in \left[0, \frac{1}{\lambda(1-q)}\right) \text{ for } q<1$ |

pdf = $(2-q) \lambda e_q(-\lambda x)$ |

cdf = $1-e_{q'}^{-\lambda x / q'} \text{ where } q' = \frac{1}{2-q}$ |

mean = $\frac{1}{\lambda (3-2q)} \text{ for } q < \frac{3}{2}$
Otherwise undefined|

median = $\frac{-q' \ln_{q'}(1/2)}{\lambda} \text{ where } q' = \frac{1}{2-q}$ |

mode =0|

variance = $\frac{q-2}{(2q-3)^2 (3q-4) \lambda^2} \text{ for } q < \frac{4}{3}$ |

skewness = $\frac{2}{5-4q} \sqrt{\frac{3q-4}{q-2}} \text{ for } q < \frac{5}{4}$ |

kurtosis = $6\frac{-4q^3 + 17q^2 - 20q + 6} {(q-2)(4q-5)(5q-6)} \text{ for } q < \frac{6}{5}$ |

entropy =|

mgf =|

cf =|

}}

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.Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356 The exponential distribution is recovered as $q \rightarrow 1.$

Originally proposed by the statisticians George Box and David Cox in 1964,{{cite journal |last1=Box |first1=George E. P. |author-link=George E. P. Box |last2=Cox |first2=D. R. |author-link2=David Cox (statistician) |title=An analysis of transformations |journal=Journal of the Royal Statistical Society, Series B |volume=26 |issue=2 |pages=211–252 |year=1964 |mr=192611 |jstor=2984418 }} and known as the reverse Box–Cox transformation for $q=1-\lambda,$ a particular case of power transform in statistics.

Characterization

=Probability density function=

The q-exponential distribution has the probability density function

: $(2-q) \lambda e_q(-\lambda x)$

where

: $e_q(x) = [1+(1-q)x]^{1/(1-q)}$

is the q-exponential if {{nowrap|q ≠ 1}}. When {{nowrap|1=q = 1}}, e_q(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

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

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:

: $\alpha = \frac{2-q}{q-1} ,\quad \lambda_\mathrm{Lomax} = \frac{1}{\lambda (q-1)}.$

As the Lomax distribution is a shifted version of the Pareto distribution, the q-exponential is a shifted reparameterized generalization of the Pareto. When {{nowrap|q > 1}}, the q-exponential is equivalent to the Pareto shifted to have support starting at zero. Specifically, if

: $X \sim \operatorname{\mathit{q}-Exp}(q,\lambda) \text{ and }
Y \sim \left[\operatorname{Pareto}\left(x_m = \frac{1}{\lambda (q-1)}, \alpha = \frac{2-q}{q-1}\right) -x_m\right],$

then $X \sim Y.$

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

: $X = \frac{-q' \ln_{q'}(U)}{\lambda} \sim \operatorname{\mathit{q}-Exp}(q,\lambda)$

where $\ln_{q'}$ is the q-logarithm and $q' = \frac{1}{2-q}.$

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.{{cite journal|title=Modelling train delays with q-exponential functions|author=Keith Briggs and Christian Beck|journal=Physica A|year=2007|volume=378|issue=2|pages=498–504|doi=10.1016/j.physa.2006.11.084|arxiv=physics/0611097|bibcode=2007PhyA..378..498B|s2cid=107475}}

It is also found in atomic physics and quantum optics, for example processes of molecular condensate creation via transition through the Feshbach resonance.{{cite journal|doi=10.1103/PhysRevA.94.033808|title=Landau-Zener extension of the Tavis-Cummings model: Structure of the solution|author1=C. Sun|author2=N. A. Sinitsyn |journal=Phys. Rev. A|volume=94|issue=3|year=2016|pages=033808|bibcode=2016PhRvA..94c3808S|arxiv=1606.08430|s2cid=119317114}}

Notes

External links

[http://www.cscs.umich.edu/~crshalizi/notebooks/tsallis.html Tsallis Statistics, Statistical Mechanics for Non-extensive Systems and Long-Range Interactions]

Category:Statistical mechanics

Category:Continuous distributions

Category:Probability distributions with non-finite variance