# Empirical Characteristic Function

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The empirical characteristic function (ecf) of a random sample {, , ...} from a statistical distribution is defined by

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Contributed by: Bob Rimmer (March 2011)

Open content licensed under CC BY-NC-SA

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The characteristic function of a statistical distribution is the Fourier transform of the derivative of its distribution function. A stable distribution is used in the example. Stable distributions lack for most cases a distribution function with an explicit formula. The standardized stable characteristic function, however, is straightforward:

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At , , as must the empirical characteristic function. The ecf converges with the characteristic function most quickly where is close to zero. As the sample size grows, the convergence becomes closer further from .

Empirical characteristic functions can be used for parameter estimation in cases where the characteristic function of the statistical distribution is known. They can be used alone in the same way one would use a known characteristic function. For instance, the characteristic function of a sum of random variables is equal to the product of the characteristic functions of each independent random variable. Or a symmetrized characteristic function can be created from the product of the characteristic function and its conjugate. Examples and *Mathematica* code for use of empirical CharacterCode functions can be found at mathestate.

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