Q-exponential distribution

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 ${\displaystyle q\rightarrow 1.}$

q-exponential distribution

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

Originally proposed by the statisticians George Box and David Cox in 1964,^[2] and known as the reverse Box–Cox transformation for ${\displaystyle q=1-\lambda ,}$ a particular case of power transform in statistics.

Characterization

Probability density function

The q-exponential distribution has the probability density function

${\displaystyle (2-q)\lambda e_{q}(-\lambda x)}$

where

${\displaystyle e_{q}(x)=[1+(1-q)x]^{1/(1-q)}}$

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

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

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:

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

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

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

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

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

${\displaystyle X={\frac {-q'\ln _{q'}(U)}{\lambda }}\sim \operatorname {{\mathit {q}}-Exp} (q,\lambda )}$

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

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.^[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