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Q-exponential distribution
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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. The exponential distribution is recovered as
Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.
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).
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.
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:
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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. The exponential distribution is recovered as
Originally proposed by the statisticians George Box and David Cox in 1964, and known as the reverse Box–Cox transformation for a particular case of power transform in statistics.
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).
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.
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:
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