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

Selection effect — where it appears

The distortion in a quantity conditioned on having been chosen for being extreme, which is a property of the choosing rather than of the estimator. The number that chose a model is biased downwards as a report on that model by exactly the amount that made it the winner.

Named by 35 essays across 20 fields — each of them below, with the objects they name alongside it.

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.

charged · Break point What each rule gives up against an oracle that is arithmetic. Expected squared error of the candidate each rule selects, minus the expected squared error of the best candidate in the table, over 500 draws of 120 rows. Both quantities are closed forms — σ_S²(1 + q/(n − q − 1)) — so the only Monte Carlo here is over which candidate got picked. The hold-out spends half its rows measuring what the criterion computes, and pays 1.8 times as much for it. Schwarz's criterion is worst because it is answering a different question: which candidate contains the truth, rather than which one forecasts best.

A criterion is a prediction of the hold-out

A rolling hold-out spends half the sample measuring what a criterion computes from all of it. Against an oracle that is arithmetic rather than an estimate, the criterion gives up 0.01701 and the hold-out 0.03200 — and the number the hold-out reports for its own winner is optimistic by more than either.

proxy · Order-selection 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 five 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.

apiece · Order-selection The number the comparison was missing. What it costs to choose the tuning parameter for every candidate separately rather than once for the table, under AR(1) at 0.8, paired on the draw. The window's figure is the one the earlier field reported; the order's is the one it named and did not make. They are the same size — 0.00401 against 0.00360, at 2.30 and 1.72 paired standard errors — and matching the lists at eight values leaves them the same size again. The prediction that the longer list would make the order's cost the larger of the two is not what happens; what happens is that the two rules cost the same once they are scored by the same criterion, which took a missing term to arrange.

The comparison that was not made

Choosing a whitening's window separately for every candidate costs 0.00401 of regret. The same question about an order was named and left, because the two lists are different lengths. The order's answer is 0.00360, and matching the lists changes almost nothing.

lists · Order-selection The eighth was not a constant. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations between the candidates, over 800 draws apiece. The earlier field reports this flat at about an eighth across list length, on a table and a world it never varies. Vary how much the omitted coefficients are worth — one multiplier, with the table, the list, the law and the sample size all held — and it runs from 17.6% to 1.5%, a factor of 11.75. The world in which every candidate is true is the world in which the tuning list decides most; the world in which one candidate dominates is the world in which it decides nothing.

The eighth that was not a constant

How often a per-candidate tuning list changes which candidate wins is reported flat at about an eighth across list length. Vary how far apart the candidates are instead and it runs from 17.6% to 1.5%.

turnover · Order-selection Three rules and a target none of them is aimed at. Which block length each rule picks, over 400 samples of 120 rows, for the tapered window. Two of the rules are points: a length written into a protocol is 8.00 on every draw and the rule of thumb is 4.00, because n to the one third does not read the data at all. The plug-in reads the sample's own persistence and lands at 14.36 with a standard deviation of 2.93. The length that would actually have been best on that draw averages 24.57 with a standard deviation of 16.23 and runs from 10 to 48 between its tenth and ninetieth percentiles. The target moves five times as much as the best estimate of it does, which is why no rule can be close to it and why the two that do not try are not merely worse — they are somewhere else.

The length nobody has

Every comparison of block windows in this collection is made at each window's own best block length. That length has a standard deviation of sixteen across draws and averages twenty-five. No rule is aimed at it.

feasible · Bootstrap What each rung 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.

separate · Break point 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.

twice · Break point How much of one search the other has already found. Five pairs of searches on one sample, on a scale whose zero and one are both fixed by construction. Zero is two searches over disjoint sets of independent columns: they remove shares of the residual sum that add, at 0.8 standard errors from exactly additive, and they read 0.004. One is a break search paired with a step column it contains, which reads exactly one on every draw because the step adds nothing at all. Between them: two dictionaries of step columns cut a few rows apart read 0.125, and the pair the earlier field measured — a break and a whitening window, both reading the same residual series — reads 0.762, three quarters of the way to one search containing the other. And below zero, a break paired with a search over independent columns reads -0.306: the joint search finds configurations neither half of it contains, so charging the two separately under-charges.

Two searches that share nothing

Two searches over independent columns remove shares of the residual sum that add exactly. On the scale a chi-square point is quoted on they look super-additive by a fifth of a unit, and none of it is overlap.

apart · Criterion What the second search finds, alone and afterwards. For four of the pairs, what the second search removes on its own and what it removes once the first has already run. The gap between the two is the overlap in absolute terms. Where the searches share nothing the two readings are the same: an independent column removes 0.0261 alone and 0.0260 afterwards. Where one contains the other they are 0.1387 and exactly zero. The pair the earlier field measured sits between: a whitening window removes 0.5033 alone and 0.3120 after a break search has run. This is the earlier field's own reading of its pair, on the share scale rather than in log-likelihood units, and it is the number a rule that runs both searches actually has to charge for.

A search that is already the other

A break search shifts every coefficient after a row, so a step column is one of the directions it can move in. Paired with a dictionary of them it reads exactly one, on every draw, and that fixes the top of the scale.

apart · Criterion Eight candidates, one of them exactly as good as the benchmark. The candidate set: moving averages of the last 1, 2, 3, 5, 8, 13, 21 and 34 observations, each drawn as its expected squared error divided by the benchmark's — the mean of all 60. The persistence is not chosen, it is solved for: at φ = 0.4895 the best candidate in the set, the average of 2, has exactly the benchmark's expected squared error, and every other candidate is worse by between 0.5% and 5.1%. So the null that no candidate beats the benchmark is true, with one candidate on its boundary. Everything a set comparison claims about its own error rate has to be measured here, because anywhere further inside the null every procedure flatters itself.

Eight forecasters and one benchmark

A set of forecasters is a multiplicity problem on top of a dependence problem, and the two do not separate. Eight windows of one series carry the multiplicity of two and a half independent comparisons; eight separate problems carry eight.

ranking · Multiplicity 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%.

twice · Break point 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.

apiece · Order-selection A near-perfect cancellation, at one correlation. A covariate skewed at 0.75 under a lower-tail copula, at each of 7 rank correlations. The earlier field measures this cell at a Spearman of 0.4 and reads 0.002% where adding the two halves' own leaks gives 16.579% — a cancellation so near exact that it is that field's headline. Across the sweep the same cell reads 0.0084%, 0.0182%, 0.0097%, 0.0015%, 0.0864%, 0.4693%, 1.5327%. Its smallest value is at 0.4, in the interior, and by 0.7 it is 1007.40 times larger. The near-zero is where two curves cross, and they cross beside the one correlation that was measured.

The zero that was a crossing

A cell that leaks 0.002% where adding its two halves gives 16.6% is a field's headline. On a finer grid it passes through zero at a rank correlation of 0.38 — two hundredths from where it was measured.

stronger · Adjustment Two factors, opposite directions. The two factors the cost of a per-candidate tuning parameter is a product of, as the candidates are pulled apart, over 800 draws at each of 5 separations. How often the candidates disagree about the tuning parameter rises from 31.8% to 88.8%; the share of those disagreements that change which candidate the table selects falls from 51.6% to 1.7%. So the setting where the candidates quarrel most about the tuning parameter is the setting where the quarrel matters least, and a sweep that reads the rate and stops has read the factor pointing the wrong way.

Two factors pointing opposite ways

As the candidates on a table are pulled apart, they quarrel about the tuning parameter three times as often and the quarrel decides the winner thirty times less often. A sweep that reads the first factor has read the one pointing the wrong way.

turnover · Order-selection 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.

separate · Break point 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.

twice · Break point What a longer list actually changes. How often the five candidates choose different tuning parameters, at a true null where every one of them contains the truth, so a disagreement is manufactured rather than discovered. The order's list is an interval of integers, and thinning it moves the rate smoothly from 0% at two values to 39% at thirteen. The window's is not an interval — it runs 0, 1, 2, 4, 8, 12, 20, 30 — so a thinned window list jumps depending on whether it happens to keep the width the criterion wants, between 0% and 42% with no order to it. So "the same length" was never quite the same thing for the two rules, and it is a smaller effect than the field it was invoked to explain.

A list is not a rule

How often five candidates disagree about a tuning parameter runs from nothing at two values on the list to two draws in five at thirteen. What the disagreement costs does not move at all.

lists · Order-selection 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.

separate · Break point Flat along a row, apart between them. The probability that a per-candidate tuning list changes the winner, at three list lengths on three candidate tables, over 800 draws in each of the nine cells. Along a row — the reading the earlier field takes — it moves by a factor of at most 1.21, so that field's invariant survives on every table. Down a column it moves by up to 1.98. The list length is the dial that does not move this number and the table is one that does, and the earlier field varied only the first.

A table and a list

A nested ladder of candidates differing by one coefficient was predicted to turn over more often at every list length. It turns over less at every one, and its list changes the winner half as often.

turnover · Order-selection 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.

charged · Break point 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.

apiece · Order-selection The weights may not read the block they weight. A weighted least squares decomposition needs weights that are constants, or at least independent of the differences they multiply. One λ̂ pooled across the trial is estimated on hundreds of degrees of freedom and is effectively a constant; a λ̂ estimated inside each block is estimated on that block's own two or three, and is correlated with the difference it weights. Coverage falls from 94.68% to 82.76% — and the interval gets wider while doing it, 0.5163 against 0.3024, which is the signature of weights that are noise.

The condition that cannot be dropped

The weights may not read the block they weight. Estimate the variance ratio inside each block rather than across the trial and the coverage falls to 83% — on an interval that is at the same time seventy per cent wider.

corner · Allocation The row count entered twice, and a penalty is one place. Regret against the best available model as the errors are made persistent. Counting rows more than quadruples; both penalty repairs — the trace, and the scalar effective sample size — are worse than it at every persistence measured; and whitening the sample and keeping the ordinary penalty falls, recovering 86.9% of what counting rows gives up at ρ = 0.85. Doing it at an estimated ρ recovers 80.6%, so having to estimate the dependence from the rows being selected on costs 7.3% of what knowing it is worth. Mallows' forms are drawn beside the logarithmic ones and behave the same, which is what rules the linearisation out.

The repair that was exact and made it worse

A penalty computed from the trace is exactly the optimism it estimates, and selecting with it gives up a fifth more than not correcting anything. The row count entered the criterion twice, and a penalty is the second place.

effective · Forecast The ladder is the same ladder under every law. Each pair's overlap under each of the four laws, over 400 draws apiece. The scale is fixed at both ends by construction: two searches over disjoint sets of independent columns read -0.001, 0.013, -0.002, 0.004, and a break search paired with a step column it already contains reads exactly one under every law. Between them the pair that reads one residual series twice runs 0.763, 0.805, 0.577, 0.752 — lowest under long memory, where a whitening has most to do and the break search has least left to find that the whitening has not taken. The rung that moves most is the pair of step dictionaries, from -0.148 under a moving average to 0.419 under a break; and the pair that is negative is negative under all four.

Three quarters of the way to one search

The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.

apart · Criterion The argument is a third of the size of the thing it is inside. Three quantities on one scale, in points of the error in a block resample's implied long-run variance, at 120 rows. The gap between the two windows at the best available block length — the whole subject of the comparison this field inherited — is 2.12 points. What the best rule a practitioner could actually run gives up against that same best length is 7.26, a factor of 3.42. What the rule of thumb gives up is 26.01. So the ordering between windows is worth establishing and is not worth arguing about, and the sentence that follows from it is not use the taper but estimate the block length, because that is where the points are.

What choosing the length costs

The gap between two block windows at the best available length is 2.12 points. What the best rule a practitioner could run gives up against that same length is 7.26. The argument is a third of the size of the thing it is inside.

feasible · Bootstrap What a schedule is allowed to read, and what happens when it reads more. The construction allows the block sizes to be anything at all as long as they are functions of the within-block contrasts, which are independent of every block mean. A schedule that shrinks the block whenever the between-block spread is running above what the contrasts say is a direct attempt to hold down the quantity the interval will be built from, and it succeeds: the estimate lands at 0.8373σ² against the honest 0.9831, and the coverage goes with it. Reading the running mean instead pushes the other way and over-covers — which is not a repair, it is the same violation with the sign reversed, and the level is no longer a property of the procedure at all.

A schedule that reads the mean

The block sizes may be anything at all provided they are functions of the contrasts. Two natural schedules break that, in opposite directions — and the most natural mistake of the three is not a schedule at all but a stopping rule, at 86.87% coverage and fewer observations.

pace · Stopping 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.

charged · Break point One window for the table, or one each. The regret of the same fifteen-candidate table under AR(1) at 0.8 over 150 draws, with the window attached three ways. Chosen once from the fullest candidate's residuals it gives up 0.02518. Chosen from each candidate's own residuals, with the covariance estimate still shared, it gives up 0.02799 — a paired cost of 0.00281 at 2.0 standard errors for the tuning parameter alone. Estimating the covariance per candidate as well costs 0.01087, so the objection already on record is about 3.9 times the size of the one that was not.

A window for every candidate

The window and the order a whitening needs are chosen once, from the fullest candidate, on an argument that was made about an estimated covariance. A tuning parameter is not a covariance, and the two cost different amounts.

together · Order-selection Four intervals at one stopping time. 3,500 experiments under the sequential rule with a first stage of 5 and a required half-width of 0.4, which is a demand that knowing σ would meet with 24.0 observations and which the rule meets with 20.8. The rule's own interval covers 90.3%. Replacing the fixed width by a t interval on the same data gives 92.0%. Keeping the rule's own random sample size and drawing a fresh sample of that size gives 89.8% at the fixed width and 95.5% for a t interval — so the sample size being random costs nothing, and the sample size being chosen by the data the interval is built from costs the rest. The spread estimated at the stopping moment is 17.1% below the truth, which is the same fact one level down.

The interval after a stop it chose

A rule that stops when the estimated precision is good enough stops on the samples whose estimate was small. Its interval covers 90% and claims 95%, and a fresh sample of the same random size covers 95.4%.

guarantee · Stopping The same dial, on a list that steps by one. The probability that a per-candidate tuning list changes which candidate the table selects, at each of 5 separations, for both tuning parameters at a matched list length of 8. The sieve order runs from 13.1% to 2.6%, a factor of 5.00; the whitening window, from 16.4% to 1.5%, a factor of 11.75. What is held is the number of options, the table, the law, the sample size and the seeds; what cannot be held is the size of a step, since an integer step and a geometric step are different amounts of change. The dial moves both, and it moves them by 2.35 times as much on one as on the other.

A step that is not a ratio

Run the separation sweep on a tuning list of integers rather than a geometric ladder and the two factors still point opposite ways. The invariant does not survive: along a row of integers the probability moves by 2.163 where along the geometric ladder it moves by 1.208.

turnover · Order-selection The price of insurance is noise, not coverage. What a robust standard error costs under a constant error variance — the case where the model-based one is exactly right — at five sample sizes over 20000 draws at the small end. It is not coverage: the leave-one-out interval read against a t on n − 2 covers 95.52% at 20 rows against the model-based 95.06%. It is a wider interval, by a factor of 1.0689 at 20 rows falling to 1.0049 at 250, and it is a variance estimate 2.57 times as variable at 20 rows and 1.85 times at 250 — against a denominator that is exactly V²·2/(n − 2), so only the numerator is counted. The uncorrected estimate is the cheaper of the two at small samples and the dearer at large: 1.31 against 1.76.

Right for the wrong reason

A robust standard error costs no coverage where the risk is absent — 95.52% against 95.06% at twenty rows. It costs a 6.89% wider interval and a variance estimate 2.572 times as variable, and the pre-test that would avoid paying recovers 15.9% of what the insurance is worth.

sandwich · Misspecification Reading the draw changes what is charged, not what is tracked. The correlation between the band width each rule picks and the best band width on the same draw, over 400 draws. The three fixed charges read -0.069, -0.066, 0.012. The three that read the sample read -0.012, -0.019, -0.041. None of the six is distinguishable from nothing. The statistic the first plug-in reads does vary — the draw's own summed squared autocorrelation runs from 2.06 to 10.19 with a mean of 4.36 — so the failure is not that the charge stopped moving. It is that what it moves with carries no information about which width this draw wanted.

A charge that reads the draw

Three charges built to read the sample track the best band width on their own draw at −0.012, −0.019 and −0.041, deliver more error than the fixed rule they are calibrated to, and pick a width half again as variable. The statistic moves; the answer does not.

curve · Criterion Every finding Benjamini–Hochberg made in thirty families, with its interval, at real effects of 2. 85 findings, sorted by their estimate. 12 of their ordinary 95% intervals miss the true effect, every one of them on the far side; 6 of the wider false-coverage-rate intervals miss.

Intervals for the findings

Benjamini–Hochberg's findings usually go out each with its ordinary 95% interval. With ten real effects of two standard errors among twenty tests, 11.59% of those intervals miss their effect, every miss on the far side, and the interval around the most prominent finding covers 72.36% of the time — 2.38% when the effects are one standard error. Intervals widened for the number of findings hold the share that miss under 5%.

multiplicity · Multiplicity How fast a gap has to close before a sample can see it close. The power of the test against the half-life of a disagreement, at 100, 200, 400 observations, each read against its own simulated critical value. Every pair in every reading is genuinely tied together, so a non-rejection is a miss. At 200 observations a gap that halves in 3 steps is found 99.9% of the time and one that halves in 12 steps is found 15.3% of the time — and by 35 steps the reading is 6.1%, which is the test's own size. Beyond that the curves are flat because there is nothing left to detect with.

How slow a return a sample can see

At two hundred observations the test finds a gap that halves in five steps four times in five, one that halves in eight 37.3% of the time, and one that halves in fifty 4.95% of the time — which is the rate at which it finds pairs with no mechanism at all. The boundary moves with the sample, not with its square root.

timeseries · Spurious

Named alongside it

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

Monte CarloInformation criterionModel selectionDependenceSpecification searchRegretOverfittingLikelihood ratioStructural breakSupremum statisticWhiteningTuning parameter

All concepts