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

Parameter uncertainty — where it appears

The part of a prediction's error that comes from the model's coefficients having been estimated rather than known. It shrinks with the estimation sample while the model's own future noise does not, so at any useful horizon the second term dominates.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

What a 95% forecast interval covers, counted. 1200 series of 25 observations from an AR(1) with φ = 0.7, at each horizon, on one set of seeds. The upper line is the interval computed at the true parameters — it covers 95.3% on average, which is the check that σ²Σψ² is the right formula rather than a claim about anything a forecaster can do. The lower line is the same formula fed σ̂² and φ̂: 92.8% at one step and 87.3% at 6. The interval that would cover what it claims is 6.9% wider at one step.

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.

forecast · Forecast A test between nested models, under a null that is true. 1000 comparisons: an AR(1) truth, forecast by a fitted AR(1) and by a fitted AR4 whose extra coefficients are zero. In population the two forecasts are identical, so every rejection is false. The larger model's mean squared error is 1.1663 against 1.0583 — worse, by exactly the noise in estimating coefficients that are not there — and the ordinary test therefore declares the smaller model significantly better 67.2% of the time. Read one-sided in the direction anybody asks about, it finds the larger model better 0.0% of the time. Adding the squared difference between the two forecasts back into the loss differential puts the level at 4.9%.

When one model contains the other

The comparison a forecaster most often wants is between a model and the same model with one more term. That is exactly the comparison the standard test cannot make — and it fails by declaring the smaller model significantly better, more confidently the more data it is given.

evaluation · Forecast A line in the right width beats two curves. How far each candidate charge sits from the measured optimism across the plateau, in units of each width's own standard error, over 2000 draws. The straight line through the origin in the band's summed weights — which is what the earlier field levies — misses by 0.2382 per width. The same straight line in the pairs the band actually uses, Σ w(k)(1 − k/n), misses by 0.0095. A fitted power law misses by 0.0293 and a fitted decaying rate by 0.0172, both on one fitted constant more. The deferral this field answers asked for a curve; the answer is a line, in a variable with nothing fitted in it.

A line that beats two curves

A deferral asked for a curve. Fitted against the same measurements, a straight line in a variable nobody had to fit describes the plateau better than either curve does with a constant more — and for three windows out of four it does not.

curve · Criterion Least squares estimates persistence low, by an amount with a formula. 3000 series of 50 observations at each persistence. The lower curve is the counted bias of the least-squares estimate of φ, and the open marks on it are −(1 + 3φ)/n, computed rather than fitted. The upper curve is the bias left after adding that quantity back, evaluated at the estimate rather than at the truth nobody has: -0.0020 at φ = 0.3, -0.0023 at φ = 0.5, -0.0039 at φ = 0.7, -0.0059 at φ = 0.8, -0.0108 at φ = 0.9, -0.0165 at φ = 0.95. The formula is a leading-order expression and it understates the bias where the persistence is nearest one — -0.0882 counted against -0.0770 predicted at φ = 0.95, which is the corner of the parameter space every one of these approximations is worst in.

Correcting the persistence

Least squares estimates how much a series remembers of itself as smaller than it is, at every value it can take, by an amount with a closed form. Subtracting that amount back is one line of arithmetic, and what the line costs is variance.

evaluation · Bias A filled value is not an observation. What a 95% interval for the slope actually covers after each way of handling 35.0% missing outcomes, counted over 4000 studies of 200 rows. Dropping the incomplete rows covers 95.93%. Filling with the observed mean covers 13.85%, because the estimate itself has moved. Filling with a fitted value covers 80.85% against a closed prediction of 79.73%: the estimate is right and the reported standard error is short by a factor of 0.6567 against a predicted 0.6500, because the residual sum of squares is divided by the whole sample's degrees of freedom. Adding residual noise recovers the spread and covers 85.78% against a predicted 84.62%, since the interval still ignores the variance of having imputed at all.

One imputation is not an observation

Three ways of filling a missing outcome, under a mechanism that makes dropping the rows beyond reproach. Filling with the observed mean covers 13.85%, filling with a fitted value covers 80.85%, adding noise covers 85.78%, and the thing all three were meant to improve on covers 95.93%.

missing · Missingness What the interval covers once the order is chosen as well. 1200 series of 40 observations from an AR(1) with φ = 0.7, at each horizon, on one set of seeds. The upper line is the interval computed at the true parameters — it covers 95.5% on average, which is the check that σ²Σψ² is the right formula rather than a claim about anything a forecaster can do. The lower line is the same formula fed σ̂² and φ̂: 93.6% at one step and 90.8% at 6. The third line chooses the order by AIC from the same data before computing the interval, which costs a further 0.8 points at h = 6.

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.

forecast · Order selection The correction does not arrive at the truth, it passes it. The average decay factor a forecast applies to the last observation, at φ = 0.85 and 50 observations, 3000 series per horizon. The middle curve is φʰ, what the model actually does. Below it is the uncorrected forecast, which uses φ̂ʰ and reverts too fast — 24.8% short at h = 4, 30.0% short at h = 6, 32.7% short at h = 8. Above it is the forecast built on the corrected estimate, which overshoots, and the reason is arithmetic rather than a bad correction: raising an unbiased estimate to a power does not give an unbiased estimate of the power, and the higher the power the more the spread of φ̂ is converted into overshoot.

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.

evaluation · Bias The extra 1/m, and the correction nobody quotes. What a pooled 95% interval covers against the number of imputations, counted over 2000 studies of 200 rows at 35.0% of outcomes missing. Rubin's rules — total variance W̄ + (1 + 1/m)B, read against a t distribution on (m − 1)(1 + W̄/((1 + 1/m)B))² degrees of freedom — cover 94.10% at two imputations and reach their promise by 5, at 95.25%. Dropping the (1 + 1/m) factor takes two imputations to 93.10%; using a normal quantile instead of the degrees-of-freedom correction takes it to 92.55%; dropping both takes it to 91.45%. The median degrees of freedom at two imputations is 12.95, which is why the second correction is the larger.

The variance between imputations

Pooling several filled datasets covers 94.10% at two imputations and reaches its promise at five, where a single fill covered 85.78%. The correction everybody quotes is the smaller of the two doing the work — 1.00 ± 0.22 points against 1.55 ± 0.28.

missing · Missingness The damage does not stay in the term that was left out. Where each coefficient lands when the model that fills the missing outcomes and the model that analyses them disagree, over 1500 studies of 200 rows at 35.0% missing and 20 imputations. An imputer that omits a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing fraction — -0.1405 counted against a closed -0.1400 — and pushes the coefficient it did impute on the other way by exactly the product of the omitted coefficient, the covariates' correlation and the missing fraction: 0.0402 counted against 0.0420. Both closed forms come out of the same two-by-two solve. Matching models leave both alone, and so does an imputer that knows more than the analysis.

An imputation model the analysis does not contain

A model that fills the gaps without a covariate the analysis fits attenuates that covariate's coefficient by exactly the missing share, 0.4 to 0.26, and moves the one it did carry by exactly γρf, 0.6 to 0.642. The reverse case is supposed to inflate the interval, and at four strengths of the extra knowledge it does not.

missing · Missingness How often seven 95% intervals cover the small groups of a league table whose size relation is steepening below the median size. Groups of twenty patients or fewer, 600 tables of a hundred groups. Coverage: one centre, plugged in 58.0% (average width 2.81); one centre, its uncertainty added 58.5% (average width 2.83); a fitted line, plugged in 83.1% (average width 1.58); a fitted line, its uncertainty added 88.4% (average width 1.77); a fitted quadratic, plugged in 85.5% (average width 1.53); a fitted quadratic, its uncertainty added 93.7% (average width 1.92); the true relation and spread 94.8% (average width 1.55).

The interval a small group is given

A league table's 95% interval for a small group, pooled towards a size relation fitted from the table, covers 92.8% of the time when the relation is straight and 83.1% when it steepens below the median — and 77.4% for the groups of eight patients or fewer. Adding the fitted centre's own uncertainty, weighted by how much the group borrows from it, brings the line to 88.4% and a fitted quadratic to 93.7%, and the quadratic's interval covers 94.6% of the smallest groups at every shape tried. One centre, plugged in, covers the smallest groups 35.2% of the time when the relation steepens, and no added variance repairs a centre in the wrong place.

borrowed · Shrinkage What the forecast interval is short by, φ = 0.85, 6 steps ahead. The plug-in interval covers 88.42% against a claimed 95%. Correcting the variance recovers 0.56 points, propagating the persistence's own standard error recovers 0.40, correcting the persistence recovers 2.66, and all three together recover 4.20 — leaving 2.38 points unaccounted for.

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.

evaluation · Bias Every group's 90% rank interval in one league table whose size relation is steepening below the median size, with its true rank. A hundred groups from four to four hundred patients, ranked from their posterior under a fitted quadratic size relation whose uncertainty is drawn once per table. The groups of twenty patients or fewer have rank intervals 30.1 places wide on average, and 34 of their 36 true ranks fall inside; the groups of over a hundred, 32.1 places, with 27 of 30 inside.

A ranking with its uncertainty

A league table's reader wants a rank, and a rank interval inherits a fitted size relation's errors differently from an effect interval. When the relation steepens below the median size, a fitted line's 95% interval for a small group's effect covers 83.1% of the time; its 90% interval for the same group's rank covers 95.6%, because the line's error lifts the small groups into the crowded middle of the table, where rank intervals are wide. The error surfaces instead in the large groups' ranks, 87.6%, and in width: 36.9 places against the 27.3 the true relation needs. A quadratic's uncertainty has to be drawn once per table, since it moves every small group together — added group by group it over-covers at 92.8%.

borrowed · Shrinkage Five 95% intervals for an AR(1) forecast 6 steps ahead at φ = 0.85, fifty observations. Coverage on 4,000 series, the same series for every interval, each simulated interval built from 499 simulated futures. The plug-in formula: 88.48%. The formula, all three repairs: 92.80%. Simulated: parameters drawn: 92.55%. Simulated: residual bootstrap: 88.25%. Simulated: bootstrap, bias removed: 93.47%.

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%.

evaluation · Bias How often a league table's own data choose a bent centre over a straight one. Over 400 tables of a hundred groups each. The quadratic's curvature coefficient is significant at 5% on 5.3% of tables whose relation is straight, 14.5% of tables whose relation is flattening below the median, 29.3% of tables whose relation is steepening below the median; its average AIC weight is 0.374, 0.456, 0.558.

A centre chosen by its own table

A league table that picks its centre's shape from its own data — a line unless a curvature test says otherwise — picks the line on 85.5% of tables whose relation flattens below the median, because the bend lives where the groups are smallest and noisiest. Its bottom ten then flags like the line's: 2.61 innocent small groups a table against 2.19 for a quadratic told the shape. Always fitting the quadratic costs 0.07 of a correctly flagged group when the relation is straight and matches the told answer when it bends, and a flag set by probability carries the choice through a mixture of the two.

borrowed · Shrinkage

Named alongside it

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

Plug in estimateMonte CarloLeast squaresCoverageForecast intervalStationarityBias correctionForecast horizonMean squared errorModel selectionClosed formConfidence interval

All concepts