Q-exponential distribution

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q-exponential distribution
Probability density function
Parameters q<2 shape (real)
\lambda >0 rate (real)
Support x\in [0;+\infty )\!{\text{ for }}q\geq 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.

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.

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 _{{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 {\mbox{qExp}}(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

Economics (econophysics)

The q-exponential distribution has been used to describe the distribution of wealth (assets) between individuals.[2]

See also

Notes

  1. ↑ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
  2. ↑ Adrian A. Dragulescu Applications of physics to economics and finance: Money, income, wealth, and the stock market arXiv:cond-mat/0307341v2

Further reading

External links

 
Continuous univariate with support whose type varies
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