Edgeworth series

An Edgeworth expansion refines the central limit theorem by adding corrections determined by higher cumulants. Let $X_1,X_2,\dots$ be independent and identically distributed with mean zero, variance ${\sigma }^2>0$ and skewness ${\gamma }_1=\mathbb{E}{X}_1^3/{\sigma }^3$. For the standardized sum $Z_n=(X_1+\cdots +X_n)/(\sigma \sqrt{n})$, the first density correction isHere $\varphi$ is the standard normal density. This is an asymptotic statement for a sequence, not a universal density formula for one arbitrary random variable. Sufficient smoothness and moment conditions are needed; one strong sufficient setting is that the common probability density function is a non-degenerate Schwartz function. In that setting the displayed remainder is valid for each fixed $z$. Lattice-valued variables need additional corrections rather than this density formula.^[ref]

To identify the coefficient, expand the log characteristic function of the standardized sum:for fixed $t$. Exponentiation and Fourier inversion turn $(it{)}^3{e}^{-t^2/2}$ into the Hermite polynomial ${\mathrm{He}}_3(z)=z^3-3z$ times $\varphi (z)$. The smoothness assumptions justify inversion beyond this formal calculation. Higher-order terms use further cumulants and Hermite polynomials.

The approximation improves central asymptotics, but it need not be a nonnegative normalized density after truncation. In particular the cubic correction can become negative in a far tail, where a fixed-point expansion does not provide a small relative error. Inverting a distribution-function Edgeworth expansion produces the Cornish-Fisher expansion for fixed quantiles.

For a standardized iid sum with a regular Edgeworth expansion, the first distribution correction is proportional to $\varphi (t)(t^2-1)$ and is even. Subtracting the two endpoint distribution functions therefore cancels the $n^{-1/2}$ contribution. Fixed symmetric pivot cutoffs can consequently have $O(n^{-1})$ coverage error even though each one-sided approximation has error $O(n^{-1/2})$. This cancellation does not automatically hold for asymmetric or estimated random cutoffs.

The Cornish-Fisher expansion inverts an Edgeworth expansion to approximate quantiles. For a standardized iid sum, let ${\gamma }_1$ be the skewness of one summand and $z_{\alpha }$ the standard normal quantile. The first distribution function correction is $-{\gamma }_1(z^2-1)\varphi (z)/(6\sqrt{n})$. Substituting $y_{\alpha }=z_{\alpha }+c/\sqrt{n}$ and expanding $\Phi$ gives $c={\gamma }_1(z_{\alpha }^2-1)/6$. This is a fixed-quantile asymptotic, with $\alpha$ fixed in $(0,1)$.