= Symmetric Edgeworth coverage cancellation
{title2=$F_n(z)-F_n(-z)=2\Phi(z)-1+O(n^{-1})$}
For a standardized iid sum with a regular <Edgeworth expansion>, the first distribution correction is proportional to $\phi(t)(t^2-1)$ 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.
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