Concept

Cumulant generating function — where it appears

The logarithm of a random variable's moment generating function, log E[exp(tX)]. Its derivatives at zero are the cumulants, and a saddlepoint approximation needs it as a whole function rather than just its first few terms.

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

Also named here as exponential tilting, saddlepoint approximation — the same set of essays touches all of them, so they are one junction rather than several.

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.

expansion · Rate

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.

expansion · Rate

Named alongside it

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

Exponential tilting Normal approximation Saddlepoint approximation Tail probability Approximation error Bootstrap Discreteness Edgeworth expansion Sample size Skewness

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