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

The rate and the size of a disagreement

A sweep of what it costs to let every candidate choose its own tuning parameter reported a product and called it a cost. Separated — and the decomposition is exact, because a draw on which nothing disagreed carries exactly zero — the rate rises by half across the list and what a disagreement is worth does not move at all. A third quantity explains why: a disagreement costs something only when it changes which candidate the table selects, that happens on about an eighth of draws, and the eighth does not depend on the list.
One factor moves and the other does not. The two factors of the same average, each drawn against its own largest value so that they share an axis. The rate at which the fifteen candidates disagree about the tuning parameter rises from 28.6% at 4 values on the list to 43.3% at 8, a factor of 1.52. What a disagreement costs, given that there was one, is 0.00975 ± 0.00224 and 0.00848 ± 0.00113 at the same two points — 0.5 standard errors apart, and the paired comparison on the draws that disagree under both lists puts it the other way. The guess this field was written to test was that a longer list makes disagreements commoner and each one smaller. The first half is right and there is no second half.

A rate times a size

A sweep reported what it costs to let every candidate choose its own tuning parameter and found it flat across the list. It was reporting a product, and the two things multiplied together do not behave the same way at all.

 What a disagreement costs, split on whether it decided anything. The regret from choosing the tuning parameter per candidate, on the draws where the candidates disagreed, split on whether the disagreement changed which candidate the table selects. Over 1200 draws at each list length: when the winner changes the regret is 0.02215, 0.03029, 0.03145; when it does not it is -0.00243, -0.00069, -0.00065 — negative, and small enough that it is inside two standard errors of nothing at every length. The whole of the cost lives in the first column, and the second column is not merely small but slightly the wrong sign: when the table's answer is unaffected, letting each candidate use its own window is a very slightly better rule than making them share one. So a disagreement about the tuning parameter is not a cost. A disagreement that changes the winner is.

The quarrel that changes the winner

A disagreement about the tuning parameter costs 0.031 when it changes which candidate the table selects and −0.0007 when it does not. The distance between the values disagreed about has nothing to do with it.

 The quantity that does not depend on the list. The probability that letting each candidate choose its own tuning parameter changes which candidate the table selects — the product of the two moving shares — against the length of the list, over 1200 draws apiece. It is 14.2%, 11.9%, 12.3%: a spread of 2.2% across a list length that moves the disagreement rate by a factor of 1.52. This is the invariant the whole field turns on. Everything downstream of the winner — the coefficients, the regret, whatever a reader is going to quote — is a function of whether the winner changed, and how often that happens is not something the list controls. A longer list changes how often the candidates quarrel and not how often the quarrel matters.

How often it matters

The disagreement rate rises by half across the list and the share of disagreements that decide anything falls by nearly the same factor. Their product — how often the tuning list changes which candidate wins — sits at an eighth and does not move.

 Which factor the sample settles. The field's decomposition at the full tuning list, by sample size, over 1,200 draws at each: how often fifteen candidates choose different values, what share of those disagreements change which candidate the table selects, and the product — how often the list changes the winner. The window: 120 rows 43.3%, 28.5%, 12.33%; 240 rows 17.5%, 14.8%, 2.58%; 480 rows 38.3%, 2.6%, 1.00%; 960 rows 11.6%, 2.2%, 0.25%. The order: 120 rows 48.9%, 21.6%, 10.58%; 240 rows 47.5%, 13.2%, 6.25%; 480 rows 46.1%, 5.1%, 2.33%.

What a longer sample settles

At a hundred and twenty rows, letting each candidate choose its own whitening changes which of fifteen candidates wins on about an eighth of draws, whatever the list, the table or the law. At four hundred and eighty rows it changes the winner on 1.0% of draws for a window and 2.3% for an order, and at nine hundred and sixty on a quarter of one per cent. The sample settles it through one factor only: how often the candidates disagree about the tuning value hardly moves, and the share of their disagreements that change the winner collapses.

 When a longer sample settles the tuning list. How often the tuning list changes which candidate the table selects, against the sample size, for the whitening window (solid) and the sieve order (dashed), on three tables of candidates, twelve hundred draws at each size. Every subset of four predictors: window 12.33% at 120, 2.58% at 240, 1.00% at 480, 0.25% at 960; order 10.58% at 120, 6.25% at 240, 2.33% at 480. The nested sequence of four: window 5.83% at 120, 1.25% at 240, 0.67% at 480, 0.25% at 960; order 5.83% at 120, 4.00% at 240, 2.25% at 480. Six subsets of two predictors each: window 11.75% at 120, 3.75% at 240, 5.75% at 480, 2.33% at 960; order 9.42% at 120, 6.67% at 240, 4.42% at 480. The nested table starts at half the full table's rate and reaches a small one a sample size sooner; the table whose candidates all have the same dimension is still changing its winner on one draw in forty-three at 960 rows.

The table decides when the sample settles it

Letting each candidate regression choose its own whitening changes which candidate wins on about an eighth of draws at 120 rows, and a longer sample settles it — for fifteen nested-and-crossed subsets, at 0.25% by 960 rows. A nested sequence of four candidates gets there a sample size sooner, from half the rate. Six candidates that all hold two predictors never get there: at 960 rows their tuning list still changes the winner on 2.3% of draws, nine times the full table's, because a disagreement among candidates of one size goes on deciding the winner as often at 960 rows as at 240.

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