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Concept
Edgeworth series
Summary
The Gram–Charlier A series (named in honor of Jørgen Pedersen Gram and Carl Charlier), and the Edgeworth series (named in honor of Francis Ysidro Edgeworth) are series that approximate a probability distribution in terms of its cumulants. The series are the same; but, the arrangement of terms (and thus the accuracy of truncating the series) differ. The key idea of these expansions is to write the characteristic function of the distribution whose probability density function f is to be approximated in terms of the characteristic function of a distribution with known and suitable properties, and to recover f through the inverse Fourier transform.
We examine a continuous random variable. Let be the characteristic function of its distribution whose density function is f, and its cumulants. We expand in terms of a known distribution with probability density function ψ, characteristic function , and cumulants . The density ψ is generally chosen to be that of the normal distribution, but other choices are possible as well. By the definition of the cumulants, we have (see Wallace, 1958)
and
which gives the following formal identity:
By the properties of the Fourier transform, is the Fourier transform of , where D is the differential operator with respect to x. Thus, after changing with on both sides of the equation, we find for f the formal expansion
If ψ is chosen as the normal density
with mean and variance as given by f, that is, mean and variance , then the expansion becomes
since for all r > 2, as higher cumulants of the normal distribution are 0. By expanding the exponential and collecting terms according to the order of the derivatives, we arrive at the Gram–Charlier A series. Such an expansion can be written compactly in terms of Bell polynomials as
Since the n-th derivative of the Gaussian function is given in terms of Hermite polynomial as
this gives us the final expression of the Gram–Charlier A series as
Integrating the series gives us the cumulative distribution function
where is the CDF of the normal distribution.
Official source
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