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

Coefficient of variation — where it appears

A quantity's standard deviation divided by its mean, which makes a spread comparable between things measured in different units. One plus its square, computed on a set of weights, is exactly the factor by which weighting inflates the variance of a mean.

Named by 3 essays across 3 fields — each of them below, with the objects they name alongside it.

What each error is a claim about, and what the claim comes out as. Each variance estimate's average over 20000 draws, divided by the variance the slope actually has across those same draws, at 80 rows with the error variance leaning towards the edges of the design (γ = 0.8). One is a standard error that is right. The model-based estimate reads 0.6081 of the spread, so its standard error is 77.98% of the one it should report; the four robust corrections read 0.9576, 0.9821, 0.9961, 1.0362. Two further routes agree with the count and share none of its arithmetic: n times the counted variance is 4.8905 against a population sandwich of 4.9200, and the counted ratio of the two standard errors is 1.2799 against a closed form of 1.2806.

The bread and the filling

The robust standard error is not a safety margin. At one setting of the error variance it is 1.2806 times the model-based one and at another it is 0.8246 times it, and the sign of a single dial decides which.

sandwich · Misspecification Three answers to how much sample is left. What a set of inverse-probability weights leaves of the treated arm, by three routes, at six settings of the assignment rule. The integral 1/(π∫φ/e) reads the whole covariate space and falls from 0.9392 to 2.655e-3. Kish's effective size counted in samples of 600 falls only to 0.2861, because almost all of the integral's fall is in a region a sample of six hundred never draws from. And the fraction the variance of the weighted mean actually delivers is lower again — 0.1155 — because the variance is the average of one over the effective size and the effective size averaged is not the same number. At the widest overlap all three agree to 0.05%.

How many observations a weight leaves

Kish's effective sample size is exact — for an outcome whose mean does not move with the covariates the weights are built from, the studentised variance reads 1.0680 where the formula says one. For the population's own outcome the same reading is 6.769, rising to 52.497.

weights · Weighting 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

Named alongside it

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

Closed formVariance ratioAsymptotic varianceCornish–Fisher expansionCumulant generating functionDesign effectEdgeworth expansionEffective sample sizeEstimated varianceExponential tiltingExtreme weightFunctional form

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