Concept

Cornish–Fisher expansion — where it appears

A correction to a normal quantile that moves a critical value by terms carrying the statistic's skewness and kurtosis. It is the Edgeworth expansion inverted, so it corrects a threshold rather than a probability, and as a polynomial in the normal quantile it eventually turns back.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

What a corrected critical value delivers: actual size over nominal, upper tail, sum of 30 exponential draws. At a nominal 1% the upper-tail test's actual size is ×1.703 of nominal with the normal critical value, ×0.991 with one Cornish–Fisher term and ×0.998 with two. At 10⁻⁸ they are ×464, ×1.418 and ×0.932.

The same correction, inverted

One skewness term added to a tail probability keeps a sum of a hundred exponential draws within 10% of the truth out to 3.09 standard deviations. The same term used to move the critical value instead keeps the test's actual size within 10% of nominal out to 5.10 — a level of about one in six million. In the short tail it has a flaw of its own: below a level of about one in a million at ten draws the corrected critical value turns back, and a test asked for a level of 10⁻⁷ rejects 1,149 times too often.

expansion · Rate

What a corrected critical value delivers to a t statistic: actual size over nominal, upper tail, 30 exponential draws. At a nominal 1% the upper-tail t test's actual size is ×0.153 with Student's quantile, ×2.94 with one Cornish–Fisher term and ×1.51 with Hall's cubic, both using the source's skewness, and ×0.878 with Hall's cubic using each sample's own. At 10⁻⁶ they are under a hundredth of nominal, ×1.04e+4, ×50.0 and ×1.59.

The correction a t test would use

On thirty exponential draws the one skewness term that held a sum's critical value within 10% out to five standard deviations nearly triples a t test's 1% size when the source's skewness is known, and its upper critical value turns back — at a level of 0.89% on ten draws, where a test asked for one in a thousand rejects 121 times too often. Hall's monotone cubic removes the turn and gains a floor. Built from the sample's own skewness it holds the short side within 20% from 10% down to one in a hundred thousand, and leaves the long side, the one a safety limit reads, at twice its level: the samples that fail are the ones that hide their skewness.

expansion · Rate

Named alongside it

The objects these essays reach for when they reach for this one.

Critical value Edgeworth expansion Skewness Tail probability Approximation error Estimated variance Monotone transformation Normal approximation p-value Student's t Upper confidence limit

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