= Edgeworth series
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= Edgeworth expansion
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An Edgeworth expansion approximates a <probability density function> by a <normal distribution> multiplied by <Hermite polynomials> encoding higher <cumulants>. For a centered variable of <variance> $\sigma^2$ and third cumulant $\kappa_3$,
$$
p(x)=\frac{e^{-x^2/(2\sigma^2)}}{\sqrt{2\pi}\sigma}
\left[1+\frac{\kappa_3}{6\sigma^3}
\operatorname{He}_3(x/\sigma)+\cdots\right],
\qquad \operatorname{He}_3(z)=z^3-3z.
$$
The correction follows from inverse-transforming the cubic term of the <cumulant-generating function>. It is an asymptotic approximation near the bulk of the distribution; truncation can give a negative result in far tails.
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