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 → 1. {\displaystyle 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 − λ , {\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
( 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 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
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 Random deviates can be drawn using inverse transform sampling. Given a variable U that is uniformly distributed on the interval (0,1), then
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