q-exponential distribution

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q-exponential distribution
Probability density function
Probability density plots of q-exponential distributions
Parameters shape (real)
rate (real)
Support
PDF
CDF
Mean
Otherwise undefined
Median
Mode 0
Variance
Skewness
Ex. kurtosis

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 .

Originally proposed by the statisticians George Box and David Cox in 1964,[2] and known as the reverse Box–Cox transformation for , a particular case of power transform in statistics.

Characterization[edit]

Probability density function[edit]

The q-exponential distribution has the probability density function

where

is the q-exponential, if q≠1. When q=1, eq(x) is just exp(x).

Derivation[edit]

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[edit]

The q-exponential is a special case of the generalized Pareto distribution where

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:

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:

Generating random deviates[edit]

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

where is the q-logarithm and

Applications[edit]

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[edit]

Notes[edit]

  1. ^ Tsallis, C. Nonadditive entropy and nonextensive statistical mechanics-an overview after 20 years. Braz. J. Phys. 2009, 39, 337–356
  2. ^ Box, George E. P.; Cox, D. R. (1964). "An analysis of transformations". Journal of the Royal Statistical Society, Series B. 26 (2): 211–252. JSTOR 2984418. MR 192611. 

Further reading[edit]

External links[edit]