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

Conditional coverage — where it appears

Coverage computed within a stated part of the population rather than averaged over all of it. It is attainable exactly for a group of positive probability by calibrating inside that group, and unattainable in finite samples given the covariate itself without further assumptions.

Named by 7 essays across 5 fields — each of them below, with the objects they name alongside it.

Which side a 95% t interval misses on, exponential source. Both tails should be 2.5%. At 8 observations the interval falls short of the mean on 9.75% of samples and overshoots on 0.31%. At 500 they are 3.31% and 2.05%, and the total is 5.36% — which a coverage table reports as very nearly right.

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.

tails · Student One promise, kept on average and inside neither group. What each calibration scheme covers inside each of two equally common groups whose noise scales are 1 and 3, over 6000 draws with 200 calibration points. One interval for everybody covers 100.00% of the quiet group and 90.66% of the noisy one, averaging to 95.28% — and the closed form for that population says 99.9999% and 90.0001% at a half-width of 4.9346, from two normal cdfs and no simulation. Dividing by an estimated per-group scale gives 95.29% and 95.48%; calibrating separately inside each group gives 95.49% and 96.01% against a closed-form expectation of 95.4645%. Only the last of those is a guarantee rather than a repair, because the rank argument runs inside each group.

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.

conformal · Exchangeability 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.

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.

survival · Censoring The interval that over-covers when the instrument fails. Counted coverage of two nominal 95.0% intervals for the same causal effect, read off the same 2000 draws of 200 rows at each first stage. The exact Anderson–Rubin set covers 95.3% at every setting — flat, because the statistic it inverts is built from y − tβ, which contains no π at all, and is therefore the same number on the same draw whatever the instrument is worth. The conventional interval covers 99.1% at π = 0.02 and 95.6% at π = 0.6: it goes wrong at the weak end by covering too MUCH, at a median width of 7.320, because its standard error is computed from residuals taken at an estimate that has itself gone wrong. A weak instrument does not make this interval lie about its coverage; it makes it useless while telling the truth.

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.

instrument · Exclusion Six scores, one coverage. What each nonconformity score's interval covers, over 2000 draws with 200 calibration points, against the 95.0249% the rank argument promises. The column runs from 94.30% to 94.90%, a spread of 0.60% against a standard error of a difference of 0.69% — one number, six times. That includes a score aimed five units off the fit and a score that never reads the response at all, because the rank argument does not read the score either: it needs the scores exchangeable and nothing else. Every decision a modeller makes has to show up somewhere else, and the next two readings are where.

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.

conformal · Exchangeability Four 95% intervals for the mean of 30 exponential observations, split by the side they miss on. Both bars should read 2.5%. The t interval misses below the mean on 6.38% of samples. Widened until its total is exactly 5%, it misses below on 4.69% and above on 0.30%. Hall's transformation misses below on 3.31% and above on 1.96%.

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%.

tails · Student The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

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.

normal · Clt

Named alongside it

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

CoverageInterval widthCalibration setConfidence intervalConformal predictionDistribution-freeEstimated varianceExchangeabilityHeteroskedasticityMarginal coverageNonconformity scoreOvercoverage

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