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

Stationary bootstrap — where it appears

Resampling blocks of geometric length so the resampled series is stationary, which keeps the dependence a block of one observation would destroy. It trades an exactly stationary resample for a block length that is random, which is a different bias-variance choice from a fixed block.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

Four blocks, and only the ends matter. The weights a block carries, normalised so that the resample keeps the residuals' variance and only their covariances are attenuated. What separates these shapes, for everything that follows, is the value at the two ends and nothing about the middle: the first-order attenuation is −(w(0)² + w(1)²)/(2∫w²), which is -1.0000 for the rectangle, -0.3896 for the trapezoid cut off at half height, and exactly zero for both windows that reach the axis. The half-height trapezoid is in the table to be the case that separates a shape from a boundary value: it is smooth, it is tapered, and it buys none of the order the other two buy.

A block weighted inside itself

The triangle every block resample attenuates by is not a fact about blocks. It is the self-convolution of a rectangle, and a block weighted down towards its own ends has a different one — whose leading term is the squared value at the two ends and nothing else about the shape.

taper · Bootstrap A quantile is the dearer reading, everywhere. The error each rule and window delivers on the two error readings, over 400 draws. The lower pair of lines is the implied long-run variance — the instrument the earlier field uses — and the upper pair is the 95% point of the standardised resampled mean, read against the finite-sample truth of 3.889 found by simulating the law directly. The quantile costs more at every one of the eight cells: at the plug-in rule it is 59.1% against 45.3% for the taper. That is not a defect in the bootstrap; a quantile is a statement about the shape of a distribution as well as its scale, and a fixed number of resamples estimates a tail worse than a variance. What matters for the comparison is that the two orderings between the windows are not the same, which the margins figure is about.

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.

readout · Bootstrap Two instruments, two block lengths. The block length that would actually have been best on each draw, for each of the two error readings, averaged over 400 samples of 120 rows. For the rectangular window the implied long-run variance wants 18.92 and the 95% point wants 16.05; for the tapered window, 21.82 against 17.74. The quantile wants a shorter block under both windows — a ratio of 0.848 and 0.813. That is the mechanism the whole field turns on: a rule for choosing a block length is a way of guessing a target, and the two instruments do not have the same target. A rule tuned to one is systematically long for the other, and the two windows do not pay the same price for being long.

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.

readout · Bootstrap How far each reference distribution's 95% point falls short. Seven constructions on rows that repeat each other, at a block length of 5 and 200 draws, against the statistic's own 95% point of 3.0224 computed from three thousand draws of the same world. Reading down: a multiplier on every row keeps no dependence at all and is 44% short; a multiplier shared along a block keeps the triangle; a fixed block keeps the same triangle and is 8% closer, which is the pair that says a taper is not what decides this; the stationary bootstrap; the two tapered blocks, both further short than the untapered one at this block length; and errors generated from a fitted model, which is the only construction here not bounded by what the residuals report.

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.

taper · Reference Eight candidates, one of them exactly as good as the benchmark. The candidate set: moving averages of the last 1, 2, 3, 5, 8, 13, 21 and 34 observations, each drawn as its expected squared error divided by the benchmark's — the mean of all 60. The persistence is not chosen, it is solved for: at φ = 0.4895 the best candidate in the set, the average of 2, has exactly the benchmark's expected squared error, and every other candidate is worse by between 0.5% and 5.1%. So the null that no candidate beats the benchmark is true, with one candidate on its boundary. Everything a set comparison claims about its own error rate has to be measured here, because anywhere further inside the null every procedure flatters itself.

Eight forecasters and one benchmark

A set of forecasters is a multiplicity problem on top of a dependence problem, and the two do not separate. Eight windows of one series carry the multiplicity of two and a half independent comparisons; eight separate problems carry eight.

ranking · Multiplicity Sixteen candidates nobody would have run, and what they cost. At φ = 0.65 one candidate in the original set is genuinely better than the benchmark, and the question is how often each procedure finds it. The added candidates are stale copies of the last value — read two, four, six … steps late — every one of them worse than the benchmark by at least 43%, and not one of them is ever the best candidate in a sample. The reality check goes from 35.5% to 0.0% as they are added, because its reference distribution has to assume every candidate is exactly as good as the benchmark and sixteen such assumptions is a critical value nothing reaches. The recentred version, which drops from the recentring the candidates the data has already ruled out — 15.1 of 24 of them — goes from 25.5% to 24.5%. The top line never moves: reporting the winner's own p-value cannot notice a change to a set it never looks at.

The models that were never in the running

A reference distribution for a set has to assume something about every candidate in it. Assuming that all of them are as good as the benchmark is what makes the reality check honest, and it is what sixteen hopeless candidates use to destroy it.

ranking · Multiplicity The reversal is a property of the instrument. The margin between the two block windows under each of four rules, on three readings of the same resampled means, signed so that a positive bar is the tapered window winning. On the implied long-run variance the taper wins at the best available block length and at one estimated from the sample and loses at a length written into a protocol and at the rule of thumb — which is the reversal the earlier field's whole argument turns on, at 1.48 and 4.52 points. On the 95% point a test actually reads, the taper wins at all four, by 6.13 to 7.08 points. On the coverage the interval actually delivers, the taper wins at all four again, by 2.50 to 5.75 percentage points. Two of the four rules change sign between the first reading and the other two, and the two that change are exactly the two the earlier field's recommendation is about.

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.

readout · Bootstrap What the interval actually covers. The coverage of the two-sided interval each rule and window builds, over 400 samples of 120 rows, against the 95% it promises. Not one of the eight reaches it: the best is 91.0% and the worst is 80.8%, on a promise of 95%. So the first thing this instrument says is that the choice between the two windows is a choice inside a range that is already four to fourteen points short, which neither of the other two readings can express at all. The second is the ordering: the tapered window covers better under every one of the four rules, by 5.00, 2.50, 5.75 and 4.00 points — including at a length written into a protocol and at the rule of thumb, where the implied variance says the rectangle wins.

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.

readout · Bootstrap

Named alongside it

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

Block bootstrapDependenceLong-run varianceMonte CarloReference distributionResamplingTaperingCritical valueBlock lengthEstimation errorPersistenceCoverage

All concepts