q-exponential distribution

q-exponential distribution
Probability density function
Parameters	$q < 2$ shape (real) $\lambda > 0$ rate (real)
Support	$x \in [0; +\infty)\! \text{ for }q \ge 1$ $x \in [0; {1 \over {\lambda(1-q)}}) \text{ for } q<1$
PDF	${ (2-q) \lambda e_q^{-\lambda x}}$
CDF	${ 1-e_{q'}^{-\lambda x \over q'}} \text{ where } q' = {1 \over {2-q}}$
Mean	${ 1 \over \lambda (3-2q) } \text{ for }q < {3 \over 2}$ Otherwise undefined
Median	${ {-q' \text{ ln}_{q'}({1 \over 2})} \over {\lambda}} \text{ where } q' = {1 \over {2-q}}$
Mode	0
Variance	${{ q-2 } \over { (2q-3)^2 (3q-4) \lambda^2}} \text{ for }q < {4 \over 3}$
Skewness	${2 \over {5-4q}} \sqrt{{3q-4} \over {q-2} } \text{ for }q < {5 \over 4}$
Ex. kurtosis	$6{{ -4q^3 + 17q^2 - 20q + 6 } \over { (q-2) (4q-5) (5q-6) }} \text{ for }q < {6 \over 5}$

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 $q \rightarrow 1$ .

Originally proposed by the statisticians George Box and David Cox in 1964,^[2] and known as the reverse Box–Cox transformation for $q=1-\lambda$ , 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 \over 1-q}

is the q-exponential, if q≠1. When 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 ~,~ \xi = {{q-1} \over {2-q}} ~,~ \sigma = {1 \over {\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 = { {2-q} \over {q-1}} ~,~ \lambda_\text{Lomax} = {1 \over {\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:

\text{If } X \sim \text{q-Exp}(q,\lambda) \text{ and } Y \sim \left[\text{Pareto} \left( x_m = {1 \over {\lambda (q-1)}}, \alpha = { {2-q} \over {q-1}} \right) -x_m \right], \text{ 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 = {{-q' \text{ ln}_{q'}(U)} \over \lambda} \sim \mbox{qExp}(q,\lambda)

where $\text{ln}_{q'}$ is the q-logarithm and $q' = {1 \over {2-q}}$

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.

Notes

↑ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
↑ 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 192611.

External links

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

Tsallis statistics

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

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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