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

Empirical quantile — where it appears

A quantile read off the sorted sample rather than off a fitted distribution. Which order statistic is taken decides the exact coverage of anything built on it, and taking the next one up rather than the nearest is what makes a conformal interval's guarantee hold at every size.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

The coverage is exact and it is not the nominal rate. ⌈(m+1)(1−α)⌉/(m+1) against m, the number of calibration points, at α = 0.05. It is a closed form and needs no data. It never falls below 95.0% and never reaches 1−α+1/(m+1), the two bounds the rank argument gives. It equals 95.0% exactly at 10 of the 182 sizes drawn — the sizes where (m+1)α is a whole number, which are 20 apart — and sits above it everywhere else, worst at 38 points where it is 97.4359%, or 2.4359% of coverage nobody asked for. Below 19 points there is no such order statistic and the interval is the whole line, which is where the curve starts.

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%.

conformal · Exchangeability The split decides the width. The width of the interval against the share of 200 observations spent on fitting rather than on calibrating, over 3000 draws. Spending more on the fit shrinks the residuals; spending more on calibration builds the interval at a less extreme order statistic. The two meet at 0.5, where the width is 4.0416 against 4.1603 at 0.1 and 4.3820 at 0.9. Full conformal, which spends the same 200 points on both jobs, is 3.9865 — so the whole cost of splitting is 1.38%.

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.

conformal · Exchangeability Which window each reading prefers. The difference between the tapered and the rectangular window, in points, on five readings at four rules: for the two errors a negative number favours the taper, and for the two coverages a positive one does. implied, variance: 1.5, 4.5, -1.1, -2.1. 95% point, uncentred: -6.1, -3.2, -7.1, -6.1. 95% point, centred: 0.9, 3.5, -0.8, -1.0. coverage, uncentred: 5.0, 2.5, 5.8, 4.0. coverage, centred: -1.2, -3.2, 1.0, 0.0. The two uncentred readings favour the taper throughout, and they are the only ones that disagree with the variance.

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.

readout · Bootstrap Too small breaks it and too large does not. Coverage of the weighted interval against the factor the true likelihood ratio is multiplied by, at a test population 80.0% drawn from the noisier group and 200 calibration points. The exact weight is the factor of 1 and covers 95.70%. Overstating it costs nothing: 96.13% at sixteen times too large. Understating it costs, and costs steeply below about a half — 94.93% at half, 88.37% at an eighth and 67.90% at a thirtieth. The question this answers was whether a wrong weight degrades smoothly or falls off a cliff, and the answer is that it does neither symmetrically: the curve is smooth and one-sided.

The weight that has to be estimated

A likelihood ratio sixteen times too large costs 5.5% of interval width and no coverage at all; one a thirtieth of the right size covers 67.90%. The estimate from a batch of five unlabelled covariates covers 95.10% against an exact repair's 95.30%, and the binomial says why.

conformal · Exchangeability

Named alongside it

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

Calibration setClosed formConformal predictionCoverageExchangeabilityOvercoverageInterval widthMarginal coverageMonte CarloOrder statisticAutoregressionBinomial proportion

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