q-exponential distribution

q-exponential distribution

Probability density function Probability density plots of q-exponential distributions
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. The exponential distribution is recovered as $q \rightarrow 1$.

Originally proposed by the statisticians George Box and David Cox in 1964, 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.

See also

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

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