Q exponential distribution - Alchetron, the free social encyclopedia



Parameters q < 2 {displaystyle q<2} shape (real) λ > 0 {displaystyle lambda >0} rate (real) Support x ∈ [ 0 ; + ∞ ) for q ≥ 1 {displaystyle xin [0;+infty )!{ ext{ for }}qgeq 1} x ∈ [ 0 ; 1 λ ( 1 − q ) ) for q < 1 {displaystyle xin [0;{1 over {lambda (1-q)}}){ ext{ for }}q<1} PDF ( 2 − q ) λ e q − λ x {displaystyle {(2-q)lambda e_{q}^{-lambda x}}} CDF 1 − e q ′ − λ x q ′ where q ′ = 1 2 − q {displaystyle {1-e_{q'}^{-lambda x over q'}}{ ext{ where }}q'={1 over {2-q}}} Mean 1 λ ( 3 − 2 q ) for q < 3 2 {displaystyle {1 over lambda (3-2q)}{ ext{ for }}q<{3 over 2}} Otherwise undefined Median − q ′ ln q ′ ( 1 2 ) λ where q ′ = 1 2 − q {displaystyle {{-q'{ ext{ ln}}_{q'}({1 over 2})} over {lambda }}{ ext{ where }}q'={1 over {2-q}}}

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 .

Probability density function

The q-exponential distribution has the probability density function

( 2 − q ) λ e q ( − λ x )

where

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

μ = 0 , ξ = q − 1 2 − q , σ = 1 λ ( 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 Lomax = 1 λ ( 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-Exp ( q , λ ) and Y ∼ [ Pareto ( x m = 1 λ ( q − 1 ) , α = 2 − q q − 1 ) − x m ] , then X ∼ 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 ′ ln q ′ ( U ) λ ∼ qExp ( q , λ )

where ln q ′ is the q-logarithm and q ′ = 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.

References

Q-exponential distribution Wikipedia

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