The Cornish-Fisher expansion inverts an Edgeworth expansion to approximate quantiles. For a standardized iid sum, let be the skewness of one summand and the standard normal quantile. The first distribution function correction is . Substituting and expanding gives . This is a fixed-quantile asymptotic, with fixed in .
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The Cornish–Fisher expansion is a mathematical technique used in statistics to approximate the quantiles of a probability distribution through its moments (mean, variance, skewness, and kurtosis) or its cumulants. It is particularly useful for adjusting standard normal quantiles to account for non-normality in distributions. In essence, the expansion transforms the quantiles of the standard normal distribution (which assumes a Gaussian shape) to those of a non-normal distribution by incorporating information about the distribution's shape.