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

Break point — the series

15 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · charged
  2. 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.

    part 2 · charged
  3. 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.

    part 3 · charged
  4. Two searches find some of the same luck. What each search reports on a sample with no break in it, and what the two report together, on four dependences. The dashed line is the sum of the two — what a rule charging each search separately would levy — and the two together always come in below it: 19.30, 15.44, 14.24, 26.64 short, on 100%, 99%, 99%, 100% of draws. The shortfall is not a rounding. Under AR(1) at 0.8 it is 19.30 of the 34.70 the break search manufactures on its own, which is more than half of it. Two searches over one sample are looking at the same noise, and the second one has less left to find.

    Two searches, one sample

    A searched break in a regression manufactures 34.7 of likelihood ratio where a count of coefficients says 11.1. A searched window manufactures 84.0. The two together manufacture 99.4, not 118.7.

    part 4 · twice
  5. 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.

    part 4 · charged
  6. Three charges, and only one of them is a test. What each of three thresholds does to the same decision, under AR(1) at 0.8, against the size of a genuine break in the mean at row 60. A chi-square on the 5 coefficients a split adds — 11.07 — declares a break on 73.6% of samples that have none: it is not a test at all. The break search's own 95% point, 69.6, carried into a rule that also chooses its window, fires on 0.0% of null samples and on 0.0% of samples with the largest break measured — the natural way of combining two published corrections does not lose a little power, it switches the test off. The calibrated charge, 27.2, holds 5.6% at no break and reaches 29.2% at the largest.

    The charge that is not a sum

    Charging two searches what each costs on its own is conservative, and conservative here means the test never fires. At the largest break measured it declares nothing, on every draw, while a calibrated threshold reaches 29%.

    part 5 · twice
  7. 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.

    part 5 · charged
  8. What a search costs is not a property of that search. The likelihood ratio a searched break in the regression reports, two ways, on every law. On its own — the whole rule being a split of the sample, at no whitening — it averages 34.70 under AR(1) at 0.8, against the 11.07 a chi-square on the five coefficients a split adds would use as a threshold. Inside a rule that also chooses a window from a list of eight, the same search adds only 15.40 — less than half. Most of what a break search finds under correlated errors is the correlation, and a whitening chosen from the same sample has taken it already. A charge measured for one search, carried into a rule that makes two, is not conservative in some harmless direction: it is measuring a different quantity.

    A charge that depends on the rule

    The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.

    part 6 · twice
  9. What each pair is made of. Each pair of searches, over 300 draws, split into the two effects its excess is the difference of. The overlap is what the second search loses by having the first already run at its own answer; the interaction is what the joint search finds by moving the first off it. They subtract to the excess exactly, on every draw, because the pinned supremum cancels. Two disjoint dictionaries of independent columns read an excess of 0.000011 and are made of 0.000514 and 0.000503. A break paired with a dictionary of step columns has an interaction of exactly 0 and is all overlap. And a break paired with an independent column has an overlap of -0.004395 against an interaction of 0.002364, which is what puts its excess below zero.

    Two effects in one number

    How much two searches over one sample share is measured as the net of two things — ground both of them find, and configurations only the joint search reaches. One extra supremum per draw separates them exactly.

    part 7 · separate
  10. How often two units contain a break that is there. 300 draws at each of 5 break sizes, a shift from row 60 of 120 on first-order autoregressive errors. The share of draws on which the profile's two-unit set contains the true row: at the fixed, unwhitened window 12.0% at 0.3, 19.0% at 0.6, 31.7% at 1, 51.7% at 1.5, 69.3% at 2; with the window searched 22.7% at 0.3, 30.0% at 0.6, 44.3% at 1, 63.3% at 1.5, 75.3% at 2. A 95% interval would sit at the top of the frame.

    A break that is there

    Put a real break in the regression at row 60 and the profile's two-unit set still misses it on most draws — it contains the true row on 12% of draws at a small break and 69% at a large one, without whitening; with the window searched as well, 23% to 75%. A set that would contain it on 95% of draws needs a threshold of about 19 units without whitening and about 8.5 with it, and either honest set holds most of the sample: fifty-seven of the seventy-three candidate rows at a break of one standard deviation. Searching the window does not move where the break is found.

    part 7 · twice
  11. Both halves grow; the difference does not. The control pair's two components and their difference, against how much each of its two searches can find, over 1200 draws at each dictionary size. Two disjoint sets of independent columns are additive at every size — the excess stays inside a standard error or two of zero throughout — and it is not because there is nothing there. The overlap grows from 0.000112 at two columns to 0.000870 at ten, a factor of 7.76, and the interaction grows with it, staying within a factor of two of the overlap at every size. Two searches competing for one residual sum share ground and find configurations neither has alone, in almost equal measure, and their difference is what the earlier field's scale calls zero.

    What a zero is made of

    Two disjoint dictionaries of independent columns read an excess of 0.000116 and are made of an overlap of 0.000583 and an interaction of 0.000467. The control the whole scale is anchored on reads zero because two effects cancel.

    part 8 · separate
  12. How often each set holds the break. For a shift added to the response from row 60 of 120 on first-order errors at 0.8, 300 samples at each of five sizes and 120 simulations per sample, the coverage of sets for the break row built at a threshold each method supplies. simulated at the truth: 95.0%, 93.7%, 93.7%, 95.0%, 94.0%; estimated break, true persistence: 90.0%, 88.3%, 88.3%, 89.3%, 90.3%; estimated break, persistence matched: 81.0%, 83.3%, 81.3%, 81.3%, 82.3%; estimated break and persistence: 64.0%, 66.0%, 67.7%, 70.7%, 75.7%; a fixed threshold of 21 units: 96.0%, 94.0%, 96.0%, 97.3%, 98.3%; the two-unit set: 12.3%, 19.7%, 35.0%, 49.0%, 66.7%, at shifts of 0.3, 0.6, 1, 1.5, 2. Simulated at the truth the construction is exact; simulated at what one sample can estimate it misses the true row a quarter to a third of the time.

    A set calibrated at the estimate

    A confidence set for where a regression breaks needs a threshold on the profile likelihood, and no fixed one works: the honest threshold falls as the break grows. Simulating the profile's deficit at the true break, shift and persistence gives a set that covers 93.7% to 95.0% at every break size and narrows from 67 rows to 23. Simulating it at what one sample can estimate gives a set that covers 64% to 76%, and most of the shortfall is one number: a split regression's residuals report a persistence of 0.62 where the errors have 0.8.

    part 8 · twice
  13. The split depends on the order. How much two searches share, measured both ways round, over 300 draws. Pinning the first search at its own answer and searching the second gives one overlap; pinning the second and searching the first gives another. A break paired with an independent column reads -0.004395 one way and 0.002163 the other, at 8.70 paired standard errors and on opposite sides of zero. The excess the two components subtract to is the same in both orders by construction, so what changes is only how it is attributed. There is no order-free way to say which of two searches found ground both can reach, and the two orders bracket it.

    A split that depends on the order

    Run the second search first and pin that instead, and the same draw gives a different overlap and a different interaction — with the same difference. And one pair has no second order at all.

    part 9 · separate
  14. The overlap climbs and the interaction does not. The over-charge at each quantile level split into overlap — the two searches' own quantiles less the sequential run's — and interaction — the joint search's quantile less the sequential run's — for three pairs over 1200 draws. two independent dictionaries: overlap -0.40, 0.01, 1.55, 2.44, 5.72; interaction 0.07, 0.01, 0.11, 0.06, 0.00. two step dictionaries: overlap -0.73, 8.19, 22.53, 31.02, 47.75; interaction 4.87, 4.55, 4.90, 4.75, 1.10. a break and a window: overlap 17.88, 29.52, 43.01, 53.79, 71.61; interaction 1.57, 1.29, 1.19, 3.22, 2.75, at the 50%, 75%, 90%, 95% and 99% points.

    The split at the critical value

    Split the over-charge of two searches at the joint statistic's 95% point rather than at its mean, and the two components behave differently. The interaction — what only the joint search can reach — stays within two units of its average at every level up to the 95% point. The overlap climbs from nothing to most of the over-charge, on two searches over independent columns as much as on searches that share ground. At the critical value the split's overlap has stopped meaning shared ground: it is mostly the arithmetic of adding tails, and the identity that defines it hides that.

    part 10 · separate
  15. One order's interaction climbs under long memory. The break-and-window pair's interaction — the joint search's quantile less the sequential run's — at five quantile levels, in the field's order (break first, solid) and the other (window first, dashed), under three laws, 1200 draws each. first-order autoregression: break first 1.57, 1.29, 1.19, 3.22, 2.75; window first 0.98, 1.26, 1.53, 2.55, 2.14. five-period moving average: break first 2.00, 1.88, 2.29, 2.63, 0.60; window first 0.16, 0.21, 0.26, 0.29, 0.00. long memory: break first 0.53, 0.80, 1.05, 0.46, 1.35; window first 2.01, 1.62, 4.75, 6.07, 7.85, at the 50%, 75%, 90%, 95% and 99% points. Under two of the laws, and in the field's order under all three, the interaction stays within a few units of its median. Under long memory with the window searched first it climbs from 2.01 to 7.85, against an average of 1.99; the same order's averages under the other two laws are 0.94 and 0.17.

    A tail the law gives the pair

    Split the over-charge of two searches at their critical value and the interaction — the configurations only the joint search reaches — stays near its average while the overlap climbs. That held under one law. Under a five-period moving average and under long memory it holds for every pair in the order the field reads, to within three units. In the other order, one pair acquires a tail of its own: with the window searched first and the break second, the interaction under long memory climbs from 2.0 at the median to 7.9 at the 99th percentile, four times its average.

    part 11 · separate

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