Field

Past the first term of the normal approximation

The Berry–Esseen bound guarantees how far a standardised sum can be from the normal, and on an exponential source it is 8.62 times the real worst error at every sample size — a constant set by a fair coin, whose error is a half-step, and at a hundred draws larger than the 2.5% tail it would be asked to certify. One Edgeworth term takes the normal's error at two standard deviations from 38% to 8% on ten draws, and turns negative in the short tail at every sample size, beyond a threshold that recedes only as the sixth root of n. The saddlepoint approximation is built at the threshold rather than at the mean, and reads a tail six standard deviations out to within 0.19% on five draws and within 2.2% on one.

The normal approximation's error on a sum of 100 exponential draws, under its Berry–Esseen bound. The distance between the exact distribution function and the normal one peaks at 0.0133, at z = -0.01. The Berry–Esseen bound is 0.1146, 8.62 times the real worst error, and larger than the whole 2.5% tail a two-sided test reads.

A bound written for a coin

The Berry–Esseen theorem guarantees how far a standardised sum can be from the normal, and the guarantee is true. On an exponential source it is 8.62 times the real worst error at every sample size, the worst error sits at the centre rather than in a tail, and at a hundred draws the bound is larger than the 2.5% tail it would be asked to vouch for.

The upper tail of 10 exponential draws: normal, one Edgeworth term, two Edgeworth terms, each against the exact tail. Each curve is an approximation divided by the exact gamma tail, so 1 is exact. Six standard deviations out at n = 10: the normal gives ×0.0000670, one Edgeworth term ×0.00159, two ×0.0167 and the saddlepoint ×1.0007.

A correction that goes below zero

One Edgeworth term takes the normal approximation's error at two standard deviations from 38% to 8% on ten exponential draws, and stretches the range within 10% of the truth from 1.66 to 3.09 standard deviations at a hundred. It also turns negative in the short tail at every sample size — past 3.13 standard deviations at a hundred draws and 9.83 at a hundred thousand — because the region recedes only as the sixth root of n.

The upper tail of 10 exponential draws: normal, two Edgeworth terms, saddlepoint, each against the exact tail. Each curve is an approximation divided by the exact gamma tail, so 1 is exact. Six standard deviations out at n = 10: the normal gives ×0.0000670, one Edgeworth term ×0.00159, two ×0.0167 and the saddlepoint ×1.0007.

An approximation built at the threshold

The saddlepoint approximation reads the tail of a sum of five exponential draws to within 0.19% six standard deviations out, where the normal is short by a factor of more than sixty thousand. It is within 2.2% out to ten standard deviations on a single draw, where there is nothing to average, and within 1.1% on a binomial whose expected count is one. It works because it is built where the tail is read rather than at the mean.

Reading the tail of a sum of thirty from one sample of 30, exponential source. Medians over two thousand samples. Six standard deviations out the normal reads ×6.8e-4 of the exact tail, the empirical saddlepoint ×0.061, a gamma fitted by the sample's skewness ×0.185, and a fixed exponential model ×1.000. The bars run from the tenth to the ninetieth percentile of the two readings that estimate the tail's shape.

A tail the sample never saw

Given the source, the saddlepoint reads the tail of a sum of thirty exponential draws to within a fifth of a per cent six standard deviations out. Given only a sample of thirty from that source, the same tilt reads it at a median of 0.061 of the truth, and nine samples in ten read it low. The error has moved from the approximation to the sample: thirty draws put the skewness at a median of 1.30 against a true 2, and hold nothing past their largest value.

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.

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.

A tilt for the ratio

A t statistic is a function of two means, so its saddlepoint tilts the joint law of each draw and its square. On an exponential source every such tilt is a truncated normal, and none has a coefficient of variation above one — the exponential is the edge of its own family. Integrated over the samples a tilt can reach, the saddlepoint reads a t statistic's tails on thirty draws within 2.5% from 10% to one in a million on the long side, with nothing fitted. The samples it cannot reach are a third of all samples, they sit at the centre of the statistic rather than in its tails, and a larger sample has more of them.

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