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

Paying for a search

A two-regime whitening chooses its change point by searching a profile, and then reads a criterion that counts parameters. Nothing charges for the search. Nothing can, in the usual way: under no break the two regimes share a coefficient and the break point is not identified at all, so the likelihood ratio is a supremum over a parameter that exists only under the alternative and no count of restrictions describes its distribution. Searching a hundred and twenty rows for a change point that is not there manufactures about five units of ratio where one parameter costs two — and a rule that counts the break as free splits a stationary sample on 99% of draws. A second break costs as much again, on a sample that has only ever had one.
The profile a break point is chosen from. One sample of 120 rows under a break in the persistence, fitted as two first-order regimes at every admissible break point. The maximum is at row 78, where the true break is at 60. The shaded band is every break point within two log-likelihood units of the best one — 8 of the 73 positions searched, which is 11% of the range. The horizontal line is the one-regime fit the search is compared against; the whole profile is above it, at every position, which is the point: a maximum over 73 candidates is above the null by construction and not by evidence.

A break that was looked for

A two-regime whitening finds its change point by maximising a profile, and then reads a criterion that counts parameters. Under no break there is no parameter to count, because every position describes the same model.

 What a search manufactures, law by law. The average likelihood ratio a search over 120 rows reports, on each of the four laws, over 400 draws. Three of them have no break at all and report 5.080, 4.839 and 4.748; the fourth has one and reports 8.757, so the real break is worth only 3.677 beyond what the search would have found anyway. The dashed line is 2, which is what an information criterion charges for one extra parameter. A search costs about two and a half of them, and the number is a measurement rather than a count.

What a search costs in parameters

An information criterion's penalty is an estimate of the optimism a fit carries. For a break point the optimism can be measured and cannot be counted, and it comes to about two and a half parameters.

 How often the split is taken, and by which rule. Over 400 draws on each of five laws. The first two rows have no break in them at all, the last two have one at row 60, and the middle one is a moving average. A criterion that counts a fitted two-regime model's parameters and nothing else takes the split on 99% of draws where there is no break. Counting the break point as one more parameter brings that to 67%. Charging what the search actually manufactures — 5.16 units, measured on a law with no break — brings it to 16%, and still takes the split on 61% of draws where there is one.

Choosing whether to break

Charging what the search manufactures takes a rule from splitting a stationary sample on 99% of draws to 16%. It also costs regret, because the two mistakes a rule can make are not the same size.

 What a second break adds. Over 200 draws, the likelihood ratio a search over one break point reports, and how much more a search over an ordered pair adds on top of it. Under AR(1) at 0.8, which has no break at all, the first search manufactures 5.697 and the second adds 4.278. Under a law with exactly one break — where a second one is as absent as the first was in the row above — the first search reports 9.442 and the second still adds 5.800. Searching for something that is not there costs the same whether or not something else was there to find.

A second break on a flat profile

Searching a hundred and twenty rows for one change point manufactures five units of likelihood. Searching for a second manufactures four more, on a series that has at most one — and on a profile whose whole range is under seven.

 Where two profile likelihoods put the break. Under a break in the persistence at row 60 of 120, and the same break turned over, over 300 draws: how often each search's estimate lands within twelve rows of the truth, with its average and spread beside it. persistent first · one scale for the whole series: 46.7%, at 74.8 with a spread of 11.8; persistent first · a scale per regime: 94.3%, at 61.8 with a spread of 5.8; persistent last · one scale for the whole series: 49.3%, at 45.3 with a spread of 11.7; persistent last · a scale per regime: 93.0%, at 56.7 with a spread of 5.9.

The break a shared scale hid

The break searches in this field all maximised a likelihood with one innovation variance for the whole series, and on a break from 0.95 to 0.65 they found it badly: an average of 74.8 where the truth is 60, within twelve rows on 47% of draws, from a profile everyone called flat. Give each regime its own innovation variance — one more parameter, and the one the law actually has — and the same search lands at 61.8 with a spread of 5.8, within twelve rows on 94% of draws, reporting four times the likelihood. The break was never barely there; the likelihood was blind to the half of its evidence that is in the scale. Under no break the extra parameter costs about one and a half units of manufactured likelihood.

All essays