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

Ranking — where it appears

Ordering groups by an estimate and reading position as a result, as a league table does. With groups of unequal size the order depends on each group's precision as much as its effect, so a rank needs its own interval.

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

The ranking on the left, the weights on the right. Eight moving-average forecasts of an AR(1) at φ = 0.4895, the persistence at which the best of them exactly ties the 60-observation benchmark. On the left, each candidate's expected squared error in units of the series' own variance: the smallest belongs to L = 2, at 1.0156. On the right, the weight each carries in the variance-minimising combination of all eight — and the best of them carries 0.00000. The two ends of the family carry 1.0172 of the weight between them, and the combination they make is worth 0.7817, which is 23.0% below the best single forecast. Both columns are closed forms in φ. Which forecast to keep and which forecasts to use are different questions, and this is a set where the answers share nothing.

The weight that is a vector

Two forecasts have a best combination and one number describes it. Eight have a best combination too, and the vector describing it puts nothing at all on the forecast with the smallest mean squared error.

search · Rank How many of a league table's top ten are small groups, ranked four ways. A hundred groups with sizes from 4 to 400, of which 36% have twenty units or fewer. Small groups make up 36.3% of the true top ten, 62.1% of the top ten by raw means, 13.4% by posterior means and 22.9% by the posterior chance of being in the top ten. The three rankings recover 4.43, 5.38 and 5.47 of the true top ten.

A league table of a hundred

A hundred groups with sizes from 4 to 400, and a top ten to publish. Ranked by their own means, small groups fill 62.1% of the top ten against their 36.3% share of the true top ten. Ranked by posterior means they fill 13.4%. The ranking built from each group's chance of being in the top ten recovers 5.47 of the true ten, the best of three and barely half; and the group ranked first could hold any rank from 1 to 31.

borrowed · Shrinkage

Named alongside it

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

ShrinkageAutocorrelationBenchmark forecastCombination weightCovariance matrixEstimation errorForecast combinationForecast encompassingHierarchical modelLeague tableMean squared errorModel selection

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