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Just a note

In other words, expanding the absolute difference between a statistic's distribution function, and a normal distribution function, (known as a Berry-Esséen rate) requires certain assumptions about how the statistic is distributed, and what sort of estimator it is. Extensions have been developed which allow for non-studentized or discrete functions, such as the binomial distribution - and for some robust estimators (such as U-statistics). But non-linear functions, or estimates based upon moderate samples of asymptotically pivotal statistics, can present problems - as do functions of contaminated and heavily-tied populations.

Provided the assumptions are met, the asymptotic difference can be expanded into a rather complicated (additive) series, known as a Chebyshev-Hermite polynomial - the constants of which (known as cumulants) are a polynomial in the moments of your statistic's distribution. The most heavily-used way of going about this, known as the Edgeworth expansion of the Cornish Fisher series, assumes you are using a normal approximation - and, following a fair amount of mathematical rearrangement, it can cope with cumulative functions and percentiles, probability densities, and inverse functions.

However, comparatively little work has been invested in other asymptotic distributions, such as the Chi-squared.