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Empirical Characteristic Function

The empirical characteristic function (ecf) of a random sample {, , ...} from a statistical distribution is defined by
.
In this representation, each random variable can be envisioned as a particle orbiting the unit circle in the complex plane. The ecf is the expected orbit or mean of the random variable orbits. For large , the ecf converges to the distribution characteristic function. The graphic shows the orbit of a standardized stable distribution with parameters and in blue. The orbit of the ecf of 500 random variables with the same parameters is shown in red and the position, at , of each random variable on the unit circle is shown as a blue dot. The red dot is the mean of these positions. Each time you change the or slider a new random sample is generated. When or , the distribution will be symmetric about zero and the characteristic function will be confined to the real line, the axis in this Demonstration.

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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:
.
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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