Coverage — where it appears
Named by 135 essays across 41 fields — each of them below, with the objects they name alongside it.
A block size that changes
The blinded rule's exactness never needed the blocks to be the same size. Letting the size be chosen from the contrasts as the run goes on leaves the coverage exactly where it was — and runs straight into an identity that says what a schedule can and cannot buy.
A width promised for a difference
The exact fixed-width interval was built for one mean. Two arms make the target 42.7 units of effective size and each unit costs four observations, so the same promise about a difference costs 169.4 rather than 42.7 — and the theorem survives untouched with the harmonic size in place of the block size.
A width the trial has to stop for
The weighting that covers at 94.9% on twelve blocks covers at 91.5% when the trial stops as soon as its interval is short enough — and so does the rule that is told every block's true variance ratio. The shortfall is the stopping, not the weights.
An interval that carries its scale
A percentile interval inherits the resampled distribution's skewness and its scale error together. The standard repair is one extra variance per resample. It was named and not run, so this runs it.
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.
The seed is part of the figure
Every other site in this fleet draws from a deterministic rule, so a figure either is or is not what it claims. Here the figures are samples, and a sample can be right by luck. That changes what a figure has to carry.
Weights that need only a ratio
A fixed-width interval about a difference is exact under either of two conditions and under neither in the corner. It is exact there too, and the only thing it needs is how much larger one arm's variance is than the other's.
What the 95% refers to
An interval that claims 95% is making a checkable statement about a procedure, not about the interval in front of you. Build every possible sample and count, and the interval taught first turns out to cover 87.6% of the time.
What a prior is worth
A prior is not a philosophical position, it is a component with a stated size. For a proportion it is worth exactly a + b observations, which turns "how much does the prior matter" from an argument into a subtraction.
What the plug-in forgets
The shrinkage weight needs a population spread, and the population spread has to be estimated from eight numbers. Empirical Bayes estimates it, substitutes it, and proceeds as though it were known — and the interval that comes out covers 79% rather than the 95% it claims.
Coverage from exchangeability alone
A conformal interval's coverage is a fact about the ranks of m+1 numbers, so it can be enumerated before any data arrive — all 40,320 orderings of eight values, agreeing with the closed form to machine precision. What that exactness delivers is not 95%.
A p-value that is not flat is not a p-value
Under a true null, p-values are uniform. That is stronger than saying the test rejects 5% of the time, it constrains the whole distribution rather than one point of it, and it catches implementation errors that a rejection rate sails past.
Two standard deviations of what
The 95.45% inside two standard deviations is a fact about a curve whose centre and width are given. Drawn from ten observations, the same band holds 91.1% on average and less than 95% on 59.9% of samples — and the average is the reading that hides it.
Where the two tails disagree
A 95% t interval on an exponential source at 120 observations covers 94.81%, which reads as very nearly right. It misses below the mean on 4.08% of samples and above on 1.11% — one tail 63% too heavy and the other 56% too light, and the total is the statistic that hides it.
A hole no sample size fills
Wilson's interval is the recommended repair for a proportion, and away from the boundary it wobbles a point or two around 95%. Near zero it has a hole: at an expected count of 0.1765 its coverage is 83.50% at ten trials, 83.79% at a hundred and 83.81% at a thousand, and it never climbs past e to the minus 0.1765, which is 83.82%. The hole is where the interval built on one success stops containing the truth, and it belongs to the count rather than to the sample size.
Five times in six
A 95% interval is read as a 95% chance that a replication's estimate will land inside it. With the spread known and a replication of the same size, the chance is 83.42% — five times in six — because both estimates are uncertain. An original that landed two standard errors from the truth captures a replication 48.40% of the time; a replication a tenth the size lands inside 44.54% of the time; and among significant originals from studies with 17% power, 66.94%.
Blinded, and still exact
The one number the exact interval needs is a ratio of within-arm spreads, which is a contrast and contains no mean — so a rule forbidden to look at the effect may compute it, on more degrees of freedom than the interval itself has.
More data is not monotonically better
Coverage of an interval for a proportion does not improve smoothly as the sample grows. It oscillates, and there are larger samples that cover materially worse than smaller ones — a sample of twenty covers twelve points worse than a sample of nineteen.
Stopping on the arms
The width a trial will report is predictable from quantities the interval is not about. A rule that stops on the prediction covers at 94.5% where one that stops on the interval covers at 91.5, and it costs two blocks and half of the width promise.
The interval that forgets it estimated
The forecast band is derived for a model whose parameters are known, and then computed by putting estimates into it. Counted, the 95% interval covers 87.3% six steps ahead on twenty-five observations, and the point forecast inside it returns to the mean a third faster than the series does.
![No block size is best at both things the procedure claims. Two claims and one dial. The honest interval's half-width falls as the blocks get smaller, because the interval's degrees of freedom are the number of blocks: 0.2602 at blocks of two against 0.2933 at blocks of sixteen. The fixed-width claim — that the mean is within 0.25 of the truth — gets more reliable as they get larger, because the sample size is less variable: 92.40% against 95.00%. Both are computed from the same runs, and the second is reproduced to within a tenth of a point by E[2Φ(d√N/σ) − 1], which needs the sample-size distribution and nothing else. The schedules sit at the bottom left: as narrow as the smallest fixed block and as few observations, with the rule's spread estimate on half as many degrees of freedom again.](/f/no-block-size-is-best-at-both-things-the-procedure-claims-c75edaf0.svg)
Two degrees of freedom, one total
The block size is a dial, and the two things a fixed-width procedure claims move in opposite directions along it. Divide the width by the square root of the sample size and one of them turns out to depend on the number of blocks and on nothing else.
Two routes to every number
A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree.
What a credible interval covers
A credible interval makes the statement everyone wants and does not claim to have a coverage. It has one anyway, it can be summed over the sample space exactly, and on a reasonable prior it beats the interval taught first.
What studentising costs
Averaged over eight cells the studentised interval is 2.09 times as wide as the percentile one and covers 0.46 points better. At the block lengths the rules choose, the scale it divides by rests on two or three numbers.
Which weights are the inverse variances
There is an exact estimator when the two arms share a variance and another when every block has the same two counts, and between them they cover every trial anybody designs on purpose. In the corner where neither holds, both cover 98.45% instead of 95%, and the only estimator at its level is the one with no theorem behind it.
What the split costs
Splitting a sample between fitting and calibrating looks like a trade against the guarantee, and it is not: coverage moves 0.63 points across nine splits and every reading sits on its own promise. The whole cost is 1.38% of width — and at sixty observations the width falls, rises and falls again.
A tenth as wide, and both of them right
The interval for a mean and the interval for one future observation are both labelled 95%, and at a hundred observations one is 10.05 times the other — exactly the square root of n + 1. Read the narrow one as the wide one and it covers a new value 15.7% of the time.
The plot is about the wrong quantity
A t interval needs the sampling distribution of the mean to be normal, not the data. A two-lump source leaves its quantile band on 100% of samples of forty and its interval covers 94.80%; a t on three degrees of freedom leaves it on 57% and covers 95.73%, the best of five sources.
What a guaranteed minimum costs
Clopper–Pearson's interval never covers less than 95%, and at thirty trials it averages 97.34% and is 10.4% wider than Wilson's. Blaker's interval keeps the same guarantee, averages 96.31% and is 4.6% wider. The difference is not waste: Clopper–Pearson guarantees each side separately, holding both below 2.5%, and Blaker guarantees only their sum — so at ten trials and a proportion of 0.15 it misses on one side 5.00% of the time.
Stopping when it is precise enough
An experiment that runs until its estimate is precise enough is the natural design and the one with a theorem against it. Its two-stage cousin keeps its promise exactly, for every unknown spread, and pays twice the observations for it.
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.
The degrees of freedom in the sums
One arm partitions N − 1 exactly. Two arms give the rule N − 2b and the interval b − 1, which is short by one per block — and the missing ones are in the block sums, which are correlated with the differences at −0.79 and are usable anyway.
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 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 rule that cannot see the mean
A sequential rule stops when its own estimate of the spread is small, which is more often on the samples whose spread came out low — so the interval afterwards is short. There is a way to keep updating the estimate and stop being able to see the mean at all.
Twenty intervals and one expected miss
The 95% belongs to the procedure, not to the interval in front of you. Twenty intervals from twenty samples make that visible in a way no definition does, and the one that misses is not a mistake.
What a schedule actually buys
Big blocks early and small blocks late is the right instinct and it does not take both ends of the trade, because there are not two ends to take. What it does take is the overshoot — about four per cent of the observations — and a steadier stopping point.
![A normal mean with a flat prior — one interval, two readings. Both intervals are [1.878, 3.446]. The frequentist reading is that the procedure captures the truth 95% of the time; the Bayesian reading is that the parameter is in this interval with probability 0.95. The endpoints are identical to machine precision.](/f/a-normal-mean-with-a-flat-prior-one-interval-two-readings-feabb882.svg)
Where the two schools agree
With a flat prior on a normal mean, the credible interval and the confidence interval are the same interval, endpoint for endpoint. Knowing exactly when that stops being true is more useful than either camp's general argument.
Marginal is not conditional
One exactly valid interval covers 100.00% of a quiet group and 90.66% of a noisy one, and the floor is arithmetic rather than a measurement — a group of share π is guaranteed only 1 − α/π, which is zero when the group is as rare as the miss rate.
Robust is not free
A robust standard error's promise is asymptotic and its use is not. Its 95% interval covers 88.73% at twenty rows, and under mild heteroskedasticity it is the worse of the two intervals until a hundred.
![One cohort of 40, and two intervals around the end of its curve. A single simulated study of 40 subjects with exponential survival at rate 0.35, dropout at rate 0.15 and follow-up to 6 — the first seed from 8811 upward whose plain band reaches below −0.05, chosen to show the failure rather than its frequency. The step curve is Kaplan–Meier and the smooth curve the truth. The plain band, the estimate plus and minus 1.96 Greenwood standard errors, first dips below zero at t = 3.78 and reaches −0.052; early on it also rises to 1.023, above one. At t = 5 the estimate is 0.069 with 1 subject still under observation, the plain interval runs from −0.052 to 0.191 and the log-log interval from 0.006 to 0.251, against a truth of 0.174. The log-log band is built on a scale that cannot leave [0, 1], and it bends away from the edge rather than through it.](/f/one-cohort-of-40-and-two-intervals-around-the-end-of-its-cdf719f3.svg)
The interval at the end of the curve
The interval most software prints around a survival curve covers 89.7% at five years, where 3.3 of forty subjects are still being watched and where the curve is actually read. The same variance carried on a log–log scale covers 94.8% there — and the failure was never the width.
A coverage table with its own error
Twenty cells estimating the coverage of an interval that is exactly 95%, at a thousand replications each, read from 93.9% to 96.5% — and a table like that flags at least one of its correct cells on 69.9% of honest runs. Ten times the replications does not repair it: at ten thousand the same table still flags one 63.3% of the time.
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.
Ninety-three observations, and nothing assumed
The interval between the smallest and largest of a sample holds a share of the population whose distribution does not depend on the population — Beta(n − 1, 2), for anything continuous. Buying the 95/95 that normality buys at ten observations costs 93 of them, and that number is the exchange rate between an assumption and data.
The coin that makes it exact
Every interval for a proportion either covers less than 95% somewhere or more than 95% on average, because a count is discrete. One construction covers exactly 95% at every proportion: it adds a uniform random draw to the count. At thirty trials it is 0.9% wider than Wilson's interval and narrower than both exact ones — and two analysts with the same data report different intervals, and one study in forty that sees nothing reports an empty one.
A block at every starting row
The t interval on the block count reads its scale from the disjoint blocks, three of them when 120 rows are cut into blocks of thirty-two, and throws away the other eighty-six blocks the series contains. Reading a block at every starting row leaves the scale's attenuation where it was — the two expectations agree to within a hundredth at every cell — and nearly doubles the degrees of freedom it realises, so the interval loses a third of its width at the same coverage. Where the interval falls short, at short blocks, it falls short by the same amount however the blocks are read.
A ratio that changes between blocks
A wrong weight costs width and a random weight costs level. The rule aimed at the quantity that actually varies is the only one that misses its own coverage, and the rule that models it across blocks recovers the whole of what knowing it is worth.
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.
The check before the standard error
One number decides whether every interval in an analysis is trustworthy, and the check for it flags a lag-one correlation of 0.5 nine times in ten — and one of 0.2 only one time in five, where the interval already covers 88.6% instead of 95%.
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%.
The interval after the choice
Estimating the coefficients of a known model costs a 95% forecast interval about two points of coverage. Choosing which coefficients to estimate, from the same forty observations, costs another four and a half — so the step nobody records in the output is the more expensive of the two.
The interval that integrates
A credible interval for one group in a hierarchy has to average over every value the population spread might take. That averaging is what makes it cover — 95.2% against the plug-in's 78.8% — and it costs 31% more width, a heavier tail, and a mixture rather than a normal.
The optimum is a ratio, and its interval is sometimes the whole line
The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.
The repair that moves the wrong number
Correcting the bias in a persistence parameter is one line of arithmetic that works. Feeding the corrected estimate into a forecast repairs the number everybody looks at, makes the forecast worse by squared error at moderate persistence, and improves the interval for a reason that has nothing to do with bias.
The shortest interval is the one that misses
Four intervals for the same data, with their widths and their coverage measured together. The narrowest is the one that fails its stated level, which is exactly why it looks the most appealing.
Two levels at once
A third level of grouping adds no new arithmetic and produces one number — the design effect — that decides how many independent observations a clustered study is worth. It is the same quantity the time-series field computes for autocorrelated data, arrived at from a completely different picture.
What a design chosen from the data costs
Two earlier measurements found what happens when a rule reads the data, and the error rate broke both times. A design that reads the data to decide where to put its runs breaks nothing — and the control that proves it also finds what the real shortfall is.
What a two-arm rule may not pool
A spread computed "within the block" without the arm label carries a share of the effect, so the trial runs 173 observations at a null and 282 at an effect of 1.5. The stopping rule is reading the thing it exists to measure, and the phrase that produced it is one word long.
What the blindfold costs
The exactly-covering rule pays for it in the width of the interval, and the block size is a dial between two costs that run in opposite directions. And on an interval whose width was fixed in advance, the same repair buys nothing at all.
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 count or the length
A block length and a block count are one number read two ways at one sample size. Read at three, the studentised interval's width penalty tracks the count — with an R² of 0.9911 against a closed form that has no length in it — and its coverage tracks the length.
The score is the modelling
Six nonconformity scores on the same draws cover within 0.60 points of each other, against a standard error of a difference of 0.69 — one number six times. Their widths run over a factor of 2.361 and their adaptivity over a factor of 8.377.
The same draws for both methods
Two intervals computed on the same simulated datasets give a difference in coverage whose variance can be 4.891 times smaller than on separate datasets — or, for a pair that covers different samples, 1.164 times larger. Which one a comparison gets is an exact sum over the counts each interval covers, and a standard error that ignores the sharing covers 100.00% for one pair and 93.07% for the other.
The outcomes a trial could have stopped with
A trial that stops at its second look with z = 3.3 has a two-sided p-value of 0.000969, 0.000987, 0.00187 or 0.0421, depending on how the outcomes it could have stopped with are ordered. One of the four orderings does not change when the looks the trial never reached are replanned, and the same one gives a trial that ran to the end with z = 6 a p-value of 0.0256.
A width rule on skewed outcomes
The blinded fixed-width rule rests on a within-arm spread being independent of the arm means, which only normal samples guarantee. On outcomes with a skewness of 4.75 the independence fails and the overall coverage barely notices — 93.60% to 94.70% across every shape counted, against 94.05% on normal outcomes. What skew moves is the runs that stop by twelve blocks, which cover about 90% with the skew in one arm, and the trial's length: a variance ratio corrected on normal theory lengthens it from 18.1 blocks to 26.0 with the skew in the first arm and shortens it to 14.2 with the skew in the second.
The side a bound is read from
On thirty exponential observations the upper limit of a 95% t interval is exceeded by the true mean 6.38% of the time, against the 2.5% a safety margin set from it assumes. Widen the interval until its total coverage is exactly 95% and the upper limit is still exceeded 4.69% of the time. A symmetric repair fixes the number that is reported and not the one that is used; Hall's transformation, which bends the interval, takes the same rate to 3.31%.
An upper limit is the finding
A survey of a thousand people with a 90%-sensitive, 95%-specific test, estimating a prevalence of one in a thousand, prints a corrected 95% interval that covers the truth 95.17% of the time — and reaches below zero in 97.95% of surveys and lies entirely below zero in 3.35%. Coverage is not the problem. The problem is what the interval can say, and the honest summary of a typical survey is one number: the prevalence is below 1.64%. Bringing that limit down to twice the truth takes 184,521 people at this specificity and 5,083 with a test that never gives a false positive.
The coin that is already there
The randomised interval covers exactly 95% at every proportion because it adds a drawn coin to the count, and a drawn coin is why nobody reports it. A sequence of trials already holds one: given the count, the order the successes arrived in is equally likely to be any of its arrangements whatever the proportion is. Used as the coin, it keeps twenty trials' coverage within 0.12 points of 95% from proportions of 0.2 to 0.8, where mid-p strays by 2.8, and it gives the same interval to every analyst. It fails where the count has few arrangements — a count of none has one — and it holds only while the order is read as recorded: an analyst choosing among sixteen orders lifts a 5% test on sixteen trials to 7.68%.
A length chosen for the interval
The squared-error rule for a block length weighs the scale's bias against its variance. Rebuilt for a block at every starting row, whose variance is two thirds as large, it asks for blocks fourteen per cent longer and buys four tenths of a point of coverage. A t interval has already paid for the scale's variance in its multiplier, so its own coverage is predicted by the bias alone — within seven tenths of a point up to blocks of twenty-eight rows — and a rule that chooses the shortest length that prediction says will cover asks for blocks two and a half to three times longer and lands within about a point of 95% at 120, 240 and 480 rows. The price is width, and it is paid where the series is short.
A ratio that changes with the period
The corner's exact interval needs one variance ratio, and the corner's own alternating allocation makes two kinds of period that need not share one. When they differ the pooled ratio is precise and wrong: at a fourfold difference the interval covers 100.0% one way round and 85.1% the other. A ratio estimated per period is exact again, blind to the effect, and costs under a tenth of the width — and a ratio that merely wanders at random costs width and no coverage at all.
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.
![Where the bias lands. The drift in the log variance ratio, fitted across 12 blocks over 4000 trials. E[log λ̂_b] is log λ_b plus ψ(k_B/2) − log(k_B/2) − ψ(k_A/2) + log(k_A/2), which depends on nothing but the degrees of freedom — so the tempting sentence is that it goes into the intercept and leaves the slope alone. It does not, because the blocks alternate between allocations and the alternation is correlated with the covariate being fitted: the lopsided blocks carry 0.5383 of bias and the even ones carry none. Uncorrected the slope reads 1.5597 against a truth of 1.5, which is 8.0 standard errors. Subtracting the two digammas block by block leaves 1.4976.](/f/where-the-bias-lands-8e6a3598.svg)
The bias that lands in the slope
The bias in a log variance estimate depends on nothing but its degrees of freedom, so it goes into the intercept — unless the degrees of freedom alternate with the design, which is exactly what a block-randomised trial makes them do.
![Where the bootstrap works and where it does not. Uniform data on [0, 1]. For the mean the percentile bootstrap covers 93.5%. For the maximum it covers 0.0%, because a resample can never contain a value larger than the largest one observed, so the interval cannot reach above it.](/f/where-the-bootstrap-works-and-where-it-does-not-943c8dea.svg)
Where the bootstrap lies
Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing.
Two intervals for one return level
Two 95% intervals read off the same fits of the same records, against a level known in closed form. The symmetric one covers 80.3% at twenty-five blocks and reaches only 89.0% at two hundred — and 99.24% of its misses are the interval sitting entirely below the truth, which is not the endpoint anybody expects to fail.
When the order matters
Three ways of breaking exchangeability cost 4.93, 11.07 and 1.07 points of coverage, and the ordering by cost is the reverse of the ordering by how soon a test would have caught them. The departure practitioners check for is the cheapest one.
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.
A simulation that stops when it looks settled
A simulation of an interval that covers exactly 95%, checked every 250 replications for a significant departure and stopped when it finds one, flags that correct interval on 29.54% of runs. Stopped instead as soon as its estimate reaches 95%, it reports an interval that covers 94% as meeting its level on 37.21% of runs. Stopped when the estimate stops moving, it reports the right number — and has quietly chosen to run about fifteen hundred replications.
Where the derivative is zero
The delta method reads a standard error off a tangent line, and at a flat point the tangent says the spread is zero. The interval built on it for a squared mean covers 99.991% there and 85.978% one and a half standard errors away, with nearly every miss on the same side — and the law it should have used is a χ², not a normal.
The interval with no resampling in it
Replace 1.96 in a normal interval on the block-means variance with Student's t on one fewer degrees of freedom than there are whole blocks, and resample nothing. Across twenty-four cells it covers at least as often as the studentised bootstrap interval at every one, by 0.42 to 10.42 points; it is narrower wherever seven blocks or fewer are left; and at fifteen blocks of 32 it covers 95.0%, which no resampled interval on the grid reaches.
The shortest interval, and the one that does not move
Two 95% intervals come out of every posterior and they are not the same set. The shorter one is shorter by 4.86% on average and 22.41% at its best, it covers 86.72% where the other covers 95.68%, and it is not even the shortest once the parameter is written a different way.
An interval for the winner
The largest of twenty correlated analyses, reported because it cleared the family-wise threshold, carries an ordinary 95% interval that covers its true mean 75.96% of the time at an effect of two standard errors and never at all when there is no effect. An interval built on the fact that it won covers 94.90% — and it is 2.48 times as wide, excludes zero for only 14.10% of the winners, and has no lower limit at all for 14.32% of them. It restores the promise by declining to say most of what the ordinary interval said.
A band allowed a few misses
From ten observations, a band holding every one of the next hundred with 95% probability reaches 4.942 sample standard deviations either side of the mean. Allowing one of the hundred outside brings it to 4.289; allowing five, to 3.378 — the 95%-content tolerance factor, 3.382, to within half a hundredth. The tolerance interval turns out to be the limit of a promise about a share of failures. And the misses arrive together: when this band misses once, it misses again 45.4% of the time, where independent misses at the same rate would do so 12.5%.
A limit from the family it came from
On fifteen exponential observations an upper limit for the mean built from the gamma family's own exact pivot, with the shape estimated by maximum likelihood, is exceeded 4.04% of the time at a mean margin of 0.746 standard deviations. Hall's transformation is exceeded 4.61% at 1.038, and the same family limit with the shape read from the sample skewness is exceeded 17.51%. Knowing the family and estimating its one parameter well recovers most of what knowing the population buys — and the parameter is estimated well only from the logarithms.
![Coverage against the expected count of the interval whose coin is read from the arrival times. The randomised interval for a Poisson mean, with its coin read from where in the window the events fell, covers exactly 97.5% at every expected count between 0.025 and ln 40 = 3.689 and exactly 95% past it, when a count of zero is given the exact interval's [0, ln 40]; its worst is 95.00%. Given mid-p's [0, ln 20] instead, it falls to 92.50% just past 3.00. Mid-p itself reaches 91.66% at 3.03; the exact interval never falls below 95.62% and averages well above 95%.](/f/coverage-against-the-expected-count-of-the-interval-whose-coin-is-read-db99b051.svg)
The coin a clock supplies
A count observed over a window comes with the times its events arrived, and given the count those times are uniform whatever the rate is — so they are a coin with a continuum of faces at every count from one up. Used to randomise the interval for a Poisson mean, they leave exactly one count without a coin, the count of zero, and what that count is given decides everything: mid-p's limit of ln 20 leaves a hole at 92.5%, and the exact interval's ln 40 is the shortest limit that leaves none, after which the interval covers exactly 97.5% below an expected count of 3.69 and exactly 95% above it.
A weighting that does not read its own spreads
On skewed outcomes a block's spreads move with its means, so a weighting fitted to every block's spreads leans: +0.0255 at a skewness of 4.75. The obvious repair — fit each block's weight with that block left out — does not remove the lean. It reverses it and nearly doubles it, to −0.0455, because a weighted mean divides by the sum of the weights and every other block's weight still reads the one left out. Splitting the trial into two halves and weighting each by the other's fit does remove it: −0.0055, indistinguishable from the weighting told the true ratios, for a spread about a tenth wider.
How much of the forbidden spread a rule may read
A stopping rule that reads the spread its interval will report breaks the interval. Blend that spread into what the rule reads, a share at a time, and the cost is not proportional to the share — at a tight requirement a rule can read four fifths of it before anything is lost, and the last fifth costs nearly four points.
The fifth a third stage never reaches
A three-stage stopping rule keeps the two-stage rule's independence and the sequential rule's budget. With a first stage of five it covers 88.4% — worse than the rule it was meant to repair — because in a fifth of experiments the second stage buys nothing.
The sums made independent
The block sums are a free second copy of a two-arm trial's variance only at balance. With ten and two in every block, pooling them covers 93.67% on four blocks; swap the arms each block and it covers 96.17%. Take from each sum the part its own difference predicts and the copy is independent at every allocation.
A share read off the sample
A block length chosen for an interval's coverage needs the share of the mean's variance the scale will find, and the published rule reads it from a first-order autoregression at the sample's lag-one value. Reading it from the sample's own autocovariances instead was supposed to survive series nobody had modelled. At 120 rows it covers 92.5% to 93.5% on all four laws — worse than the wrong shape on every one — because a centred series' autocovariances sum to exactly zero and the share they imply is too generous; read to lag forty it covers 84%. The assumed shape costs width, not coverage: 17% too wide where dependence stops at lag four, and nothing visible under a break, where the interval covers for a reason the share knows nothing about.
The interval the test inverts to
A randomisation test of no effect becomes an interval for the effect by testing every value at once: subtract the hypothesised effect from the treated outcomes and ask the same question again. Because the rule reads only covariates, one set of re-run allocations answers every value. Under minimisation the unadjusted t interval covers 99.7% at a width of 1.229; the inverted test of the same unadjusted difference covers 95.3% at 0.816, a third narrower, with no model of the outcome. Under complete randomisation the two are the same interval — which is what says the third was the rule's.
Shocks that are not normal
Every forecast interval in this field reads a normal quantile, and every series it was scored on had normal shocks. Give a first-order autoregression shocks with heavy tails and the 95% interval still covers about 95%; its 99% interval covers 98.2%. Give it skewed shocks and the 95% interval covers 94.8% one step ahead with every miss on one side — 0.0% below, 5.2% above — and the horizon does not repair it: twelve steps ahead the misses are still 0.0% below and 4.9% above, because a persistence of 0.7 keeps the error's skewness at 1.11 however far out the forecast goes. A two-sided coverage count cannot see any of this.
An interval that knows the band was maximised
A regression whose coefficients and eight-lag error band are maximised together prints an interval that covers 72% to 83% where 95% was promised, because it reports nearly the precision of a known covariance and delivers less. Two repairs were named. A sandwich standard error read off the whitened residuals repairs nothing — 71% to 83% — because the maximisation has already made those residuals look independent. A parametric bootstrap that redraws from the fitted band and maximises the band again on every redraw repairs about half the gap, to 85.5% to 89%, by pricing the maximisation through repeating it. What it cannot price is that the band it redraws from is the maximised one.
A bootstrap that repeats the design
After a design that chose its settings from the data, a parametric bootstrap can keep the settings the experiment ran or repeat the whole procedure from the start. On twelve runs repeating it changes coverage by a point at most, because the adaptivity was nearly free. What matters is the interval's shape: the percentile bootstrap covers 90.6%, no better than the Wald interval's 91.3%; the studentised one 93.0%; the profile interval 94.9%. And after a design stuck at a twelvefold wrong guess, the studentised bootstrap — the textbook repair — covers 77.8%, with intervals averaging 54.5 wide for a parameter of 3.
A ranking that belongs to the interval
Five ways of choosing a block length, read on the coverage of the interval each length builds rather than on the variance it implies. On the percentile interval the variance's verdict mostly survives: under a five-period moving average the plug-in and the whole-sequence rule tie at 120 rows and the sequence rule leads at 480. Build a studentised interval from the same resamples and the lead reverses — the plug-in's longer blocks cover 94.5% against 93.3% — because an interval that widens for its own uncertainty is paid for a long block where a percentile interval is punished for one. Under long memory no length and no interval covers two times in three.
A stopping rule that reads the periods
A fixed-width trial in the corner stops when its estimated precision reaches the promise, and it estimated that precision from spreads pooled across a trial whose two kinds of period need not share a variance ratio. Read each kind of period's own spreads instead and the trial stops on 3,019 observations where it took 10,989, at an eightfold difference, with the interval it reports still covering 95.1%. The stopping time reads nothing but within-arm spreads, per period now as well as pooled, and that is all the exactness needed.
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.
A leak read off a two-by-two table
The share of an interaction a threshold-balancing rule leaves alone needs one number from the data: how often a unit past the cut on one covariate is past it on the other. A cohort of a thousand estimates it to within a dozen points at the 90th percentile. At the 99th most cohorts have an empty cell in their table, and every one of those reads a leak of exactly one hundred per cent — a reading the bootstrap then confirms with an interval of zero width.
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.
The correction for not knowing the spread
The t distribution exists because the standard deviation is estimated rather than known. At eight observations, using the normal instead makes every interval 12% too short — and the coverage that follows can be measured rather than argued about.
The count that is not the rows
Three hundred rows in five clusters of sixty carry 6.9000 times the variance an independent-rows calculation reports, and the interval that counts rows covers 53.42%. The same five unequal sizes laid out two ways give design effects of 9.3158 and 5.4652.
A ratio whose interval has to be the whole line
The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.
The draws aimed at the tail
The chance a standard normal exceeds 5 is 2.8665×10⁻⁷, and a plain simulation needs 349 million draws to estimate it to within ten per cent. Draws aimed at the tail and weighted back need 565. Aimed slightly too narrowly, the same method has an infinite variance, an interval that covers 86.0% and gets worse with more draws, and an effective sample size that reads healthier than a proposal that works.
An interval for something else
An interval for the odds is free — put the endpoints through the odds and the coverage does not move, exactly, for any interval at all. The method everyone uses instead computes a new standard error on the new scale, and at twenty trials that costs four points of coverage, produces negative odds, and has no value at all when nothing was observed.
The level a limit should be set at
When a failure costs twenty times a standard deviation of margin, a t limit on fifteen exponential observations set at the conventional 97.5% has an expected loss of 2.143. Set at 99.95%, the same formula's loss is 1.263 — within 1% of the best any fixed multiple of s/√n can do, and ahead of both Hall's transformation and the fitted gamma family at their own best levels. Hall's is the worst at every level on every source. Choosing the level does more than choosing the construction, and the correction that reads the sample is the one no level rescues.
An interval for the analyses admitted to
An analyst runs twenty correlated analyses, admits to one, and reports the winner with an interval conditioned on having cleared 1.96 — the selection as written down rather than as it happened. The interval covers 95.3% of the time, against 95.6% for one conditioned on all twenty, because a winner's truncation point is set by the threshold far more than by its unseen rivals: hiding nineteen analyses moves it by 0.061 on average. What the hiding changes is how often there is a winner at all. With no effect anywhere, a family of twenty produces a reported winner 35.5% of the time against 4.8% when all twenty are admitted, and one whose interval excludes zero 1.48% of the time against 0.13% — the interval keeps its promise and the report does not.
Two windows and a ratio
Two counts from two windows give an interval for the ratio of their rates through the conditional binomial: given the total, the first window's count is binomial with a proportion that depends only on the ratio. The arrival times in both windows supply that binomial with a continuous coin at every total from one up, and the interval then covers exactly 95% plus a twentieth of the chance of no events at all. A total of zero can only report the whole of (0, ∞). And the exact conditional test's familiar conservatism is not the zero total's: the zero is the larger cause only while the expected total is below ln 2, and at an expected total of ten the exact interval still covers 97.8% for reasons that have nothing to do with it.
A floor that reads the first stage
A three-stage stopping rule fails at a first stage of five because a low first reading asks for nothing more, and the only floors that repair it were set near the answer, by somebody who already knew roughly what the spread was. Inflating the first reading by Stein's own factor needs no such knowledge, and it does cut the experiments left stuck from a fifth to a fifteenth. But those still stuck cover less than before, the rule needs a different dial at every requirement, and a floor of 26 observations covers 96.5% at the spend where the rule covers 92.1%, even met by a spread it was not set for.
A tail sixty residuals have barely seen
A forecast interval that reads a normal quantile has every one of its misses on one side when the shocks are skewed, and the natural repair is to draw future shocks from the model's own residuals. It works on the side: 0.0% below and 6.5% above becomes 4.3% and 4.1%. It pays for that with coverage, 93.5% to 91.6%, and on sixty observations it does this under normal shocks too. And a nominal 99% interval has a ceiling nothing in the resampling can lift, because one new shock beats all of sixty old ones in either direction with probability 3.3%.
The sums under a moving level
A trial's block sums are a free second estimate of its variance only while they see what the differences see, and a level that moves between blocks with an effect moving against it breaks that: pooled raw at a ceiling they cover 87.6% to 90.0%. Residualising the sums on an unbalanced design does not make them safe. It moves the failure. On ten and two, 90.0% at the ceiling becomes 93.8%, the trough leaves for a slope of about −1.5, and over the rest of the range the same estimate over-covers, at 99.9% on an interval half as wide again as it needs to be.
A rule that sets its own κ
A stopping rule may read the spread inside its blocks and may not read the spread between them, and the dial between the two costs nothing for a long stretch and then a lot. The share an efficiency argument recommends is each estimate's share of the degrees of freedom, and on blocks of two at a half-width of 0.15 that loses nothing and steadies the experiment by 30%. At 0.35 it loses 1.18 points. Deferring the forbidden reading until twenty blocks costs nothing measurable at any requirement, and it keeps most of the steadiness where steadiness can be had.
An effect that is not one number
A randomisation test becomes an interval for the effect by testing every constant effect in turn, so when the effect grows with the prognosis every hypothesis it tests is false. It does not fall apart. It covers the population's mean effect within a point of its level at every slope measured, and its width grows by exactly what the sample's own composition adds. A regression that models the interaction stays narrow at 0.72, covers the sample's mean at 95%, and covers the population's at 84%. The extra width was never a defect. It was the difference between two questions.
A sequence that knows it was centred
A block length read off a sample's own autocovariances gets worse the further out it reads, because a centred series' autocovariances sum to exactly zero. To first order every lag loses the same amount, about the long-run variance over the sample size, and that can be added back. Corrected, a reading to lag forty goes from 84.5% to 89.2%, and to lag twelve from 93.3% to 94.5%. But the correction overshoots on the long lags, and on the rule that picks its own cut-off, which is short, it moves coverage by three tenths of a point. The identity was the reason the lag mattered. It was not the reason the rule falls short.
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%.
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.
A standard error that knows about the instruments
Limited-information maximum likelihood came out least biased when a concentration parameter of 8 was spread over thirty-two instruments, and its conventional interval covered 79.0%. Bekker's many-instrument standard error covers 93.8% on the same draws, at 63% of the jackknife's width — and it gets there with a median standard error of 0.561 against a true spread of 0.797, because it is large on the draws that need it. At eight times the strength it covers 94.9% at 91% of the jackknife's width, and nothing measured here beats it.
When the prior is confident and wrong
A prior worth thirty-five observations, centred in the wrong place, produces a 95% interval that covers nothing at all — and reports a width 5% narrower than an honest one. It takes seventeen thousand observations to repair, not thirty-five, and the worst study to run is the one whose sample size equals the prior's weight, exactly.
What the interval is short by
The forecast interval covers 88.42% where it claims 95%. Correcting the persistence recovers 2.66 points, correcting the innovation variance 0.56, propagating the persistence's own standard error 0.40 — and all three together recover 4.20 of the 6.58, leaving a residual none of the standard repairs reaches.
The reference the sandwich is read against
The cluster-robust interval covers 75.05% at five clusters and 93.58% at eighty. The same estimate read against a t on G − 2 covers 87.95% at five, and the estimator is unchanged — three hundred rows grouped into five clusters cover 74.28% where the same three hundred grouped into seventy-five cover 94.63%.
An interval that covers and says nothing
A procedure returning the whole line 95% of the time and the empty set otherwise has coverage exactly 95% at every parameter value. Two real intervals at forty observations have expected widths of 0.2418 and 0.2417 and worst-case coverages of 55.31% and 92.21%.
The check worth more than the check
The same exactly known companion that verifies a simulation can sharpen it. On one set of four thousand draws, one companion is worth 1.08 times the draws and another is worth 214 times them, and the factor is 1 − ρ² with nothing else in it.
A flat point with more than one direction
At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.
What a two-unit study should report
The fixed-effect interval covers the mean of the two sites in hand 96.37% of the time and the population mean 54.77%. The random-effects interval covers the population mean 94.96% — exactly its level, from one degree of freedom — and is 11.6 times wider.
A winner chosen by its diagnostic
An analyst runs twenty correlated analyses and reports the one whose diagnostics look cleanest, if it clears 1.96 — and a diagnostic is rarely independent of the statistic it checks. With no effect anywhere, a diagnostic correlated 0.9 with its statistic lets a winner through in 13.07% of families against 2.52% for an unrelated one. The interval conditioned on clearing 1.96 still covers 93.27%, but its misses have moved: it excludes zero in 4.69% of reported winners, where a 95% interval should do so in 2.5%. Conditioning on the diagnostic as well repairs it, and so does something a reader can do alone — weight the reported statistic by the chance an analysis like it wins the diagnostic contest, from three numbers a report can state.
The cleanest of the significant
An analyst with twenty correlated analyses can choose in two orders: take the one whose diagnostics look cleanest and report it if it clears 1.96, or take the analyses that cleared 1.96 and report the cleanest of them. The second is the one described as prudent, because it never reports an analysis whose diagnostics look poor — and with no effect anywhere it reports a significant analysis in about 18% of families whatever the diagnostic is, against 2.55% to 13.15% for the first. The diagnostic decides which significant analysis is shown and never whether one is. Each report's own interval can still be made exact, by conditioning on a selection set that is a union of pieces rather than a half-line; the family's claim count cannot.
A count that bets against its interval
A reader who sees only the count can bet against an interval's 95%, and win whatever the proportion is, exactly when some set of counts carries less than 95% at every proportion. Out of forty trials the textbook interval has one — a count of five or fewer, or thirty-five or more, covers at most 94.90% — and the score interval has none: a prior exists under which every one of its forty-one intervals is 96.46% credible, and that prior is the proof that no bet can be found.
A proposal fitted to its own draws
The cross-entropy method fits an importance-sampling proposal from its own pilot draws, and asked to fit a normal's mean and spread to P(Z > 5) it fails before the question of variance arises: the spread halves at every stage, the level stalls near 3.7, and 95% of runs never reach the threshold, so the interval covers 1.1% of the time. Its ideal end point, the normal closest to Z conditioned past 5, is N(5.187, 0.181²), which has an infinite variance inside its own draws' reach and covers 93.7% at a thousand draws and 90.8% at ten thousand. Fitting the mean alone lands within a tenth of the best shift and covers 94.5%.
A quantile over the recent scores
A noise scale that grows across the sample takes a split conformal interval from 95% to 85.63%, and the best detector cannot see most of that loss. Take the quantile over the last thirty-nine calibration scores instead of all 199 and the worst coverage across the whole sweep is 93.42%, for 3.7% more width when nothing has drifted. The same window covers 91.30% where all 199 cover 93.53% under strongly correlated errors — and switching to it only when the detector fires is worse than either, because the detector fires on correlation too.
An effective number of clusters
Ten clusters of thirty rows, a covariate spread evenly across them, and a cluster-robust interval for the slope read against a t on G − 2 = 8 covers 91.1%. The slope needed about five degrees of freedom, not eight, because the two outermost clusters carry 49% of its variance and the middle two 0.6%. A count computed from the sizes and positions before any outcome is seen — 4.45 here, 3.32 when the big clusters sit at the edges — with the bias-reduced variance covers between 94.5% and 96.8% in all twelve layouts measured, where the usual interval on G − 2 covers between 76.2% and 94.5%.
An interval built from simulated futures
A forecast interval read off simulated futures needs no delta method and no normal multiplier, and was expected to close the residual the three repairs to the formula leave. It does not close it on its own: the plain residual bootstrap covers 88.25% six steps ahead at φ = 0.85, the plug-in formula's 88.48% to within the count, because it simulates around the same biased persistence. With that bias removed inside the bootstrap it covers 93.47%, the best of five intervals, and its shortfall stops growing with the horizon — 93.80% at twelve steps, where the repaired formula covers 91.85%.
Testing for a flat point first
Test whether the gradient is zero, and if the test does not reject, read the interval off the second-order law that holds at a stationary point. For a squared mean that two-stage rule covers about 50% at its worst whatever the pretest's level — 49.72% with a 5% test, where the tangent interval it was meant to rescue never falls below 85.98% — because its worst sits exactly where the true distance equals the test's critical value, half the samples are declared flat there, and the second-order interval cannot reach the truth from any of them.
A prior wrong about its own spread
A prior can be centred exactly where the truths lie and still be wrong about how far they spread. Ten times too confident — worth a hundred observations about proportions whose population is worth ten — its 95% credible intervals cover 48.2% of the population's studies at twenty trials, and nothing in the interval shows it: it is 0.456 of a flat prior's width, against 0.436 for a prior whose hundred observations' confidence is earned. A prior with tails too thin misses the one study in ten that comes from somewhere unusual a third of the time, and a flat component of two per cent brings that study's coverage from 65.6% to 84.0%.
A prior on the spread between two sites
Integrate the between-site spread of a two-site study under a half-normal prior of scale 1 and the interval for the overall mean is 3.01 wide and covers 94.63% when the true spread is 1 — the supplied interval again, under another name. At a true spread of 3 it covers 68.38%. Every prior tried covers 95% averaged over its own draws and none covers it at every spread; each holds up to about its own scale, because two sites move the posterior median of the spread by less than a third of what the truth moves.
Where the extremal index matters to a return level
A return level from a declustered fit depends on how often clusters arrive, which is the extremal index estimated from the same record. For a level reached once in two periods of a twenty-period record the index's error is a quarter to four fifths of the level's variance, and an interval that leaves it out covers 59.0% when clusters are long; put in, 90.0%. For a level reached once in a hundred periods its share is under two and a half per cent and adding it changes no interval at all — what fails there is the shape, fitted from however many clusters the dependence left: 82.3% coverage from forty-six clusters, 65.7% from nineteen.
Cells of unequal size and an error that follows them
Thirty-two instruments that are the indicators of cells holding 2 to 15 units each, and confounding whose variance is larger in the small cells. Limited-information maximum likelihood, the estimator that came out least biased with many instruments, then errs by 0.534 at a concentration of 8 — further than two-stage least squares — and Bekker's interval covers 74.8%. The jackknife version of the likelihood, which never lets a row's own leverage reach its own fitted value, errs by 0.066 and its robust interval covers 90.6%; at eight times the strength, 94.8% against Bekker's 87.3%. On normal instruments the same unequal variance costs almost nothing, because their leverages are nearly equal.
An interval that knows its finding was found
An interval for one Benjamini–Hochberg finding can be read from the distribution of its estimate given that it cleared the threshold it had to clear. It covers the finding's effect 94.8% of the time with real effects of two standard errors, against the ordinary interval's 87.0%, and it covers the most prominent finding 91.3% of the time where the ordinary one covers 72.4%. The price is width — a median 5.06 standard errors against 3.92 — and something larger: for three findings in four at that effect size the interval reaches back past zero, so a finding that cleared the line is, once the line is accounted for, not distinguishable from nothing.
Named alongside it
The objects these essays reach for when they reach for this one.
Confidence intervalMonte CarloInterval widthDegrees of freedomSample sizeClosed formFixed-width intervalEstimated varianceBlindingStopping ruleDependenceLong-run variance