Edgeworth expansion

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 ${\displaystyle f}$ be the characteristic function of its distribution whose density function is F, and ${\displaystyle \kappa _{r}}$ its cumulants. We expand in terms of a known distribution with probability density function Ψ, characteristic function ψ, and cumulants ${\displaystyle \gamma _{r}}$. 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)

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