Critical value — where it appears
Named by 52 essays across 28 fields — each of them below, with the objects they name alongside it.
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.
The instrument and the reading
Every comparison between two block windows in this collection is an error in an implied long-run variance. Nobody reads a long-run variance. Read on the 95% point a test uses, the same bootstrap costs half as much again.
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.
A length for each instrument
The block length that is best for an implied variance is 18.92; the one best for the 95% point of the same resamples is 16.05. A rule is a way of guessing a target, and there are two targets.
A null with a model in it
The distribution to read the winner of a table against cannot be resampled from the data, because the data does not contain the null. It has to be generated from a model — which is the assumption the resampling was chosen to avoid.
A taper and a critical value
Two constructions whose tapers visibly differ give the same critical value, and two that share a taper exactly do not. Adding a construction whose taper is a decision rather than an accident says which half of that is true.
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%.
The statistic that changes sign
A test for an unreachable half needs a quantity that tells one half from the other. Every symmetric reading of a mirror pair is identical, and a magnitude is the natural thing to reach for.
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.
Which series goes on the left
The two-step procedure has to pick a series to regress the others on, and nothing in its output records which. With a pair that choice never changes the verdict. With three series and one relation between them, the three choices disagree about whether the system is cointegrated at all 98.0% of the time.
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.
Counting what is still wandering
The statistic that turns a spectrum into an integer has one name and three distributions. Its 5% point is 8.12, 18.64 or 31.74 depending only on how many series are left wandering under the null being tested — and read against the wrong one of those three, it calls unrelated random walks cointegrated most of the time.
Dropping the losers
Carrying the best of eight arms forward and testing it at 1.96 rejects a true null 10.3% of the time — the hypothesis was chosen by looking at the data, so the statistic is a maximum wearing a single comparison's clothes. The value that holds the rate is 2.313, and it has to be solved for.
Measuring a variance rather than a quantile
A resample's implied long-run variance can be computed from the sample with no resampling in it at all. A critical value cannot, and the difference is a factor of three in the draws before any of the resampling is counted.
Residuals that keep their own variance
A reference distribution for a search has to be generated from a fitted model, and the generator draws residuals. Four ways of drawing them keep four different things — and the one this site has reached for three times repairs nothing at all here.
The ordering reverses again
One field found two of four rules changing sign between two readings of one resampling. Turn the same resamples into a studentised interval instead of a percentile one and all four change sign.
The plus one and the round number
A sampled randomisation test counts the observed allocation as one of its own reference draws, and the correction is invisible at B = 19, 39, 59 and 999 — every value anybody uses. At B = 20 the version without it is an 8.00% test where the corrected one is 3.80%, and the convention protecting everybody is a preference for round numbers minus one.
The reversal that was the instrument's
On an implied variance the rectangle wins at a protocol length and at the rule of thumb. On the 95% point a test reads, and on the coverage an interval delivers, the taper wins at all four rules.
The triangle that was not the multiplier's
A resampling that leaves each residual on its own row can keep only what the residuals have, times a triangle. A construction that moves every one of them has the same triangle — and the one in this collection's own table has a different taper entirely.
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.
Two defects and one resampling
Four resamplings, each the repair for one defect and wrong about the other. Put both defects in the same world and the statistic's 5% point is 3.8028, where the best of the four reaches 2.8326 — until a multiplier that stays on its own row and shares a sign with its neighbours reaches 2.9988.
What the first stage does not know
A single weak instrument does not make the conventional interval undercover — it makes it cover 99.1% at a width of 7.320. Where the promise actually breaks is many instruments — coverage falls from 97.2% to 51.5% while the median width falls from 1.454 to 0.583.
The trials that stopped early
A fixed-width trial that stops when its own interval is short enough covers 91.45% — an average of 78.2% among the 22.3% of runs that stop within eight blocks and 96% to 99% among those that run longer. Widening every interval by 17.1% brings the average to 95% and leaves the early stops at 85.6%, while 92.8% of runs now report an interval wider than the width they promised. Even doubling every interval leaves the early stops short.
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.
Draws that repeat each other
A hunt costs 1/p evaluations per independent draw. A walk costs one per step and yields an effective draw every τ steps. Both are counted in the same unit, and the walk is dearer at every tolerance a trial is designed at.
Errors generated from a fitted model
The one construction that is not bounded by the residuals, because a model extrapolates past the lags it was told about and a truncated sample sequence cannot. It is nearly exact where the only defect is dependence, and it pays for it where there are two.
How long a block a multiplier shares
Sharing a sign over more rows keeps more of the dependence and leaves fewer independent signs to build a distribution from. The bias falls from 1.6885 to 0.8479 and the spread rises from 1.3073 to 2.1716, and the rejection rate walks straight through its nominal level on the way from 11.3% to 1.3%.
The diagnostic at two hundred
Pointed at a trial size no enumeration reaches, the test gives three answers rather than one — and past a certain thinness it stops agreeing with itself, which is the honest reading and the one nothing could give before.
The walk that cannot cross
A thin enough admissible set is not one set. It splits into an assignment and its mirror image, no sequence of admissible single swaps joins them, and the walk that samples it is uniform on half the reference distribution for ever.
What a reference distribution costs to sample
A randomisation test on a trial too large to enumerate has to sample its reference distribution, at 1/p attempts per draw and a p-value resolved to 1/(B + 1). Six constraints cost 9,878 attempts per thousand draws, and a thousand draws resolve p to 9.99·10⁻⁴ and not one digit finer.
What the exactness buys
Against a z test calibrated to reject exactly 5% of true nulls on this design, the randomisation test loses nineteen points of power. What it buys is that the calibration needs the success rate — which moves the critical value from 1.668 to 2.718 and is the quantity the trial was run to find out.
What the interval covers
Eight rules and windows, and not one of them reaches its promised 95%. The range is 80.8% to 91.0%, and the choice between two block windows is a choice inside a shortfall that is four times larger.
The overlap at the critical value
Two searches over independent noise columns are additive on average: the joint search finds what the two find separately, to within a fortieth of the smaller. At the point a test actually uses — the joint statistic's 95% quantile — the same pair overlaps by 0.320, and a rule that adds the two searches' own critical values rejects 1.4% of true nulls instead of 5%. Nothing about the searches causes it. Quantiles of a sum do not add, and on every pair in the field the overlap at the critical value is larger than the overlap the field measured.
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.
What a multiplier cannot keep
Two reasons were named for the quarter a blocked resampling falls short, and taking either away makes the gap larger. What is left is a bound — a multiplier can only take dependence out, and the residuals' own is already below the errors'.
Two instruments that disagree
The overidentification test keeps its size at 5.0% and reaches 86.4% power against a violation carried by one instrument. Against the same error carried by both in proportion to their first stages it rejects on 4.6% of draws — its own size — while the estimate is wrong by 0.3000, which is 94.2% of the confounding the instruments were brought in to remove.
The miscalibration a perfect forecaster shows
A forecaster whose true reliability is exactly zero shows a calibration error of 0.1252 on fifty forecasts and 0.0090 on ten thousand. Every one of 1,200 blameless hundred-forecast records exceeds the 0.02 routinely read as evidence of a problem, and the mean does not fall under it until 1,976 forecasts.
The cliff that is a slope
A regression between two independent series is called significant 4.9% of the time at no persistence, 52.4% at a lag-one correlation of 0.9, and 83.4% at a unit root. The rule the field offers asks whether the last of those holds, and at 0.9 the unit-root test correctly refuses one 87.2% of the time.
The rank is a decision
The sequential procedure's 5% bounds one of its two errors. Over-counting reads between 4.2% and 7.2% at every sample length from fifty observations to three hundred; under-counting reads 69.5% at fifty and 0.0% at three hundred, and nothing in the procedure bounds it.
The same correction, inverted
One skewness term added to a tail probability keeps a sum of a hundred exponential draws within 10% of the truth out to 3.09 standard deviations. The same term used to move the critical value instead keeps the test's actual size within 10% of nominal out to 5.10 — a level of about one in six million. In the short tail it has a flaw of its own: below a level of about one in a million at ten draws the corrected critical value turns back, and a test asked for a level of 10⁻⁷ rejects 1,149 times too often.
A relation supplied in advance
When theory supplies the long-run coefficient, the gap between two series can be tested directly and the search the Engle–Granger test pays for disappears: its 5% point moves from −3.38 to −2.89, and at two hundred observations the slowest return found four times in five goes from a half-life of 5.05 steps to 6.88. The price is in what a rejection means. With a gap that halves in three steps, a coefficient a fifth too large is confirmed 99.1% of the time and one half again too large 81.9%, though every such gap contains a random walk — and at a thousand observations the second is confirmed more often, not less.
The power a calibration buys
Charging two searches separately makes a joint test conservative, and calibrating the joint statistic on its own null repairs that. Against a real break, on the pair of a break search and a window search, the repair is worth 47 points of power — 62.0% against 14.5%. On the pair of a break search and a column search it is worth nothing, and worse: a real column is found by the calibrated joint test on 11.8% of samples and by a plain Bonferroni split on 73.0%, because a joint statistic takes its threshold from the noisier of its two searches.
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.
A shrink towards the answer
This line of essays reported that the 95% point is a dearer reading than a variance, that a further point wants a shorter block, and that a taper beats a rectangle on what a test reads where a variance says the opposite. All three came from one uncentred quantile. Read about the resamples' own mean, the point costs 28.5% against the variance's 45.8%, every level wants about the same block, and the windows order alike on all readings. The taper's coverage had a separate cause: its resampled mean is 0.875 of the sample's, and every series here had a true mean of exactly zero.
A slow return across many pairs
One pair of two hundred observations is found tied four times in five only if its gap halves within 5.1 steps. Sixteen independent pairs sharing the same half-life, tested by the average of their Engle–Granger statistics, are found four times in five up to a half-life of 15.0 — and at a half-life of twelve, 99.8% of the time against one pair's 20.0%. What the widened test rejects is a statement about the panel: with half the pairs tied it rejects 55.4% of the time, with a quarter 21.2%, and a pair-by-pair reading of a panel in which all sixteen are tied finds 2.65 of them, against 0.78 when none are.
The correction a t test would use
On thirty exponential draws the one skewness term that held a sum's critical value within 10% out to five standard deviations nearly triples a t test's 1% size when the source's skewness is known, and its upper critical value turns back — at a level of 0.89% on ten draws, where a test asked for one in a thousand rejects 121 times too often. Hall's monotone cubic removes the turn and gains a floor. Built from the sample's own skewness it holds the short side within 20% from 10% down to one in a hundred thousand, and leaves the long side, the one a safety limit reads, at twice its level: the samples that fail are the ones that hide their skewness.
A horizon chosen after looking
A difference in restricted mean survival read at whichever of eleven horizons looks most convincing rejects 11.24% of trials in which the treatment does nothing, against 4.70% at a horizon fixed in advance. The correlation of the differences across horizons is closed, and the Gaussian process it defines prices the choice at a critical value of 2.317 — which brings the counted size back to 4.99% and keeps 96.92% of the power that a horizon nobody could have known to fix would have had.
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.
A detector built for the ordering
The best of three checks for a drifting scale fires at half the growth factor the standard one needs — 2.12 against 4.31 — and still leaves 6.50 points of coverage gone before it does, against 0.51 for serial correlation. The reversal was not a property of the test.
A shock every pair shares
Sixteen pairs of series whose shocks are half common are worth 5.37 independent pairs to the panel's average statistic, and sixteen that share nine tenths are worth 1.44. Read against the 5% point computed for independent pairs, the pooled test then calls a panel with no relation tied 15.5% and 31.6% of the time; read against its own point, it finds a relation shared by every pair 22.3% of the time at nine tenths common, against one pair's 20.0%. Removing each date's cross-sectional average gives back the level and all sixteen pairs when every pair feels the shock equally, and leaves 13.3% false alarms when the loadings run from a half to one and a half.
One test for a curve that might cross
The log-rank statistic and the differences in restricted mean survival at eleven horizons are one Gaussian vector under no effect, and its correlations are closed — from −0.461 at half a year to −0.949 at three, where the log-rank statistic is nearly the restricted mean itself. So one critical value, 2.374, prices the largest of all twelve. At 100 a side it keeps 93.6% of the log-rank test's power when the hazard is halved throughout and 98.5% of the best horizon's when the curves cross, where the log-rank test has none. Running both tests and keeping either rejection rejects 8.67% of trials with no effect.
A factor estimated from the panel
When sixteen pairs feel a common shock with loadings from a half to one and a half, removing the cross-sectional average leaves the pooled test rejecting an unrelated panel 13.3% of the time. Estimating the factor and each pair's loading instead brings that to 8.9% — and no further, because each loading comes from the same two hundred dates however many pairs there are — and it costs power against the average at the right critical value, 73.8% against 82.5%. Removing the shocks as observed restores 4.5% exactly and finds a shared relation 7.5% of the time: a series tied to its partner carries the partner's factor in its level, not its own shock.
Named alongside it
The objects these essays reach for when they reach for this one.
Monte CarloReference distributionStatistical powerBlock bootstrapDependenceError rateSpecification searchResamplingLikelihood ratioLong-run varianceSupremum statisticPersistence