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.

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 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%.

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.