Concept

Least favourable configuration — where it appears

The point of a composite null at which a statistic is most likely to be large, which is where a valid test of that null has to be calibrated. A table of equal-dimension candidates attains it by construction, which is what makes such a table the only one where a measured rejection rate measures the reading alone.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Sixteen candidates nobody would have run, and what they cost. At φ = 0.65 one candidate in the original set is genuinely better than the benchmark, and the question is how often each procedure finds it. The added candidates are stale copies of the last value — read two, four, six … steps late — every one of them worse than the benchmark by at least 43%, and not one of them is ever the best candidate in a sample. The reality check goes from 35.5% to 0.0% as they are added, because its reference distribution has to assume every candidate is exactly as good as the benchmark and sixteen such assumptions is a critical value nothing reaches. The recentred version, which drops from the recentring the candidates the data has already ruled out — 15.1 of 24 of them — goes from 25.5% to 24.5%. The top line never moves: reporting the winner's own p-value cannot notice a change to a set it never looks at.

The models that were never in the running

A reference distribution for a set has to assume something about every candidate in it. Assuming that all of them are as good as the benchmark is what makes the reality check honest, and it is what sixteen hopeless candidates use to destroy it.

ranking · Multiplicity

What the corner costs when the table is full of hopeless candidates. The benchmark holds two predictors, one of which is worth 1; a third predictor, worth the amount on the horizontal axis, is held only by candidates the benchmark does not contain. At the left the null is true and both procedures hold their level. To the right there is a genuinely better candidate, and the uncorrected reality check finds it 2.7% of the time while the same test with the clearly bad columns recentred finds it 51.0% of the time. The columns doing the damage are the ones nobody would have looked at twice: they are so far behind that they cannot win, and calibrating as though they might is what makes the test blind.

The corner the test is calibrated at

"No candidate is better than the benchmark" is not a null but a face of a region, and a reality check is calibrated at one corner of it. Fill the table with candidates that are hopeless rather than equal and the test finds a genuine improvement 0.0% of the time.

select · Reference

One test for eight nulls

A combination of eight forecasts can be tested against its parts one comparison at a time or all at once. All at once is a Wald test on the vector of eight encompassing means — which has seven degrees of freedom, not eight, because the means sum to zero on every origin when weighted by the combination. On sixty origins it rejects a true null 26.6% of the time, Hotelling's correction brings that to 15.1%, and handed its own true critical value it is less powerful than Bonferroni over the largest of the eight against both wrong combinations measured: 12% against 18%, and 47% against 96%.

search · Multiplicity

Power as the recentring's threshold moves. The recentred set test's power at a persistence of 0.65, where one forecaster genuinely beats the benchmark, over 300 draws of 100 origins, as its threshold runs from zero to three times Hansen's. With the eight windows alone: 30.3%, 28.3%, 26.0%, 26.0%, 26.0%, 26.0%. With sixteen stale candidates added: 30.3%, 28.3%, 24.3%, 21.0%, 19.3%, 17.7%. The studentised reality check, which sets nothing aside, has 26.0% and 17.7%.

The threshold the recentring reads

A recentred set test keeps a candidate at its own mean when the sample has placed it far enough below the benchmark, and how far is a threshold — Hansen's, about 1.75 on a hundred origins, chosen as a rate rather than tuned. Swept from zero to three times that, it is a dial with one side that costs almost nothing. At zero the test finds a genuinely better forecaster 30.3% of the time with or without sixteen stale candidates in the set; at Hansen's threshold it finds it 24.3% of the time with them, and at three times it 17.7%. Its size never passes 3.0%, even where every candidate sits exactly on the boundary.

ranking · Multiplicity

What Hansen's threshold costs, by the number of origins. The power a threshold of zero keeps over Hansen's, on the same draws. with sixteen stale candidates: 7.5%, 7.3%, 6.8%, 5.3%, 3.7%, 6.0%. the eight windows alone: 2.7%, 3.8%, 4.7%, 5.3%, 3.7%, 6.0%, at 25, 50, 100, 200, 400, 800 origins. The price does not fall as the sample grows.

A threshold against its sample

A recentred test's threshold exists so that the test is right in the limit, and at a hundred origins it costs a fifth of the power against a stale set. The natural defence is that the price falls as the sample grows and the guarantee starts to earn it. Across twenty-five to eight hundred origins it does not: a threshold of zero keeps four to seven points more power at every size, and its size at the corner stays at the level from fifty origins on. What does change with the sample is the stale set, which stops costing anything by two hundred origins under Hansen's rule and by eight hundred under the most cautious.

ranking · Multiplicity

Named alongside it

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

Composite null Monte Carlo Statistical power Benchmark forecast Familywise error rate Multiple comparisons Reality check Reference distribution Block bootstrap Bonferroni Data snooping Error rate

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