Normal Distribution
nothing is called 95% until it has been counted
Concept

Saddlepoint approximation — where it appears

An approximation to a distribution's tail built by tilting the distribution until the point read is its centre, applying the normal there, and tilting back. Its relative error barely grows with the threshold, where approximations built at the mean fail.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A tilt for the ratio, read where it exists. The saddlepoint density of the draw's mean and mean square, integrated over the samples a tilt can reach that put the t statistic past its exact critical value, over the exact tail those same samples carry, at 30 exponential draws. On the short side the relative error runs from +0.88% at 10% to +2.47% at 10⁻⁵; on the long side from +0.83% to +2.05% at 10⁻⁶. Nothing in the construction was fitted to these tails; its error stays within 2.5% of the truth at every level shown.

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.

expansion · Rate Approximate weights against the size of the run. What sixteen bands of the logarithms' t statistic are worth to a simulation of the lognormal t test's lower-tail size, in multiples of the draws, against the size of the run, with the band weights from Student's t exactly and from three approximations to it. Exact weights are worth 2.709 at every size. Fisher's one-term expansion is worth 2.708 at four thousand draws and 2.523 at a million, and breaks even with counting at 23.7 million draws. The variance-matched normal is worth 1.849 at four thousand and breaks even at 14,698. The standard normal is worth 0.762 at a thousand and breaks even at 669: a bias that does not shrink is multiplied by every draw added.

Weights from an approximate law

A companion whose law is only approximately known can still be cut into bands, with each band's weight computed from the approximation. The weight's error is a bias that does not shrink, and the run multiplies it: the stratified rate is worth p(1 − p)/(v + Nb²) times the draws. Reading Student's t on nineteen degrees of freedom as normal breaks even at 669 draws; the normal with the t's variance at about fifteen thousand; Fisher's one-term expansion at twenty-four million. What decides the bias is how the weight errors line up with the band rates, not how large they are — Fisher's is wrong by sixty per cent in one band and biases the rate by six hundredths of one.

splits · Routes

Named alongside it

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

Cumulant generating functionEdgeworth expansionExponential tiltingTail probabilityNormal approximationSkewnessStudent's tApproximation errorBias-varianceBinningBootstrapClosed form

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