A shrink towards the answer

This essay corrects three earlier ones, and it should say so first.

The instrument and the reading said the 95% point of a block bootstrap costs half as much again as its implied variance. The reversal that was the instrument’s said that a rectangular block beats a tapered one on the variance at two rules and loses on the 95% point and on coverage at all four. A length for each instrument said a quantile wants a shorter block than a variance, because a long block leaves few blocks in a resample and a tail is where that shows. Each was a careful measurement of the number it computed. The number was not always the one its text named.

The question that found it was innocent. If a tail wants a shorter block because it reads the part of the distribution furthest from normal, then a further point should want a shorter block still, and a test at 99% should run on less than a test at 90%. That was going to be the essay. It is a one-line sweep: resample at each length, read the point at each level, score each against that level’s finite-sample truth.

The answer came back backwards. The best block rose with the level, from 4.7 at the 75% point to 9.6 at the 99%. And the 75% point had a relative error of 162%, which no bootstrap of a well-behaved mean should have. That was the thing worth following.

The point nobody was reading #

The bootstrap draws resamples $x^*$ and records $\sqrt{n}\,\bar{x}^*$ for each. Its distribution is centred near $\sqrt{n}\,\bar{x}$, the sample’s own mean. The truth it was scored against is the 95% point of $\sqrt{n}\,\bar{x}$ under the law, which has no sample in it and is centred at zero. So the earlier reading compared a point sitting around $\sqrt{n}\,\bar{x}$ + 3.9 with a truth at 3.9. The sample’s mean, which has a standard deviation of about 2.4 in those units, went into the error.

That is not what a test reads. A test of a mean compares $\sqrt{n}(\bar{x} - \mu_0)$ with a critical value, and the critical value is the point of $\sqrt{n}(\bar{x}^* - \bar{x})$: the resamples about their own centre. The essay’s title said “the 95% point a test reads”, and the computation read something else. Nothing is wrong with the resampling, and no check on the arithmetic could have caught it, since such a check asks whether the reading is a quantile of the array, and it is.

The size of the mistake is simply arithmetic. At a persistence of 0.7 the error in the centred point is about 0.27 of its truth. The sample’s mean adds about $2.4 / 3.9 \approx 0.61$ of it, independent of that, so together they make $\sqrt{0.27^2 + 0.61^2} = 0.67$. The earlier essay measured 0.66.

So the first claim reverses. On a rectangular window at the protocol length the variance costs 45.83%, the uncentred point 65.51% and the point about its own mean 28.48%. At the rule of thumb the three are 56.95%, 68.17% and 35.52%. At the plug-in rule, 46.45%, 66.23% and 28.85%. At the best length in hindsight, 39.05%, 53.64% and 23.07%.

The ratio is no mystery, and it is not about tails. A point is a multiple of a scale, and the relative error of a square root is half the relative error of what is under it. 28.48 against 45.83 is 0.62, where the first-order figure is 0.50 and the rest is the concavity. A quantile is the cheaper reading, for the reason any standard deviation is easier than a variance, and the earlier essay’s sentence that a quantile “is a statement about the shape of a distribution as well as its scale” turns out to be nearly idle here, as the next section measures.

A further point does not want a shorter block #

Now the sweep can be read as it was meant to be, centred, at five levels and three persistences, on 300 draws with a thousand resamples each.

The lines are flat. At a persistence of 0.5, the five levels want 6.5, 6.7, 6.6, 6.6 and 6.4. At 0.7, they want 9.5, 9.9, 9.8, 9.7 and 9.3. At 0.9, 18.5, 18.3, 18.2, 17.9 and 17.5. Across the range there is a drift of about a block at the highest persistence, and at the lower two the differences are within what a coarse grid can resolve.

So the mechanism was not there. A resample of $n$ rows from blocks of $\ell$ is a sum of $n/\ell$ pieces, and a sum of six pieces is further from normal than a sum of thirty. That is true, and it costs almost nothing here, because the series is Gaussian and a Gaussian’s block means are Gaussian however few of them there are. So the “normality a long block destroys” was never destroyed. The tail is not where the error comes from.

The direct check is to skip the tail altogether. Read the same resamples’ standard deviation, multiply by the normal point for the level, and score that: a reading with no tail in it at all. At 0.7 it has its least error at 9.5, 9.6, 9.6, 9.6 and 9.7, with errors of 0.260, 0.264, 0.265, 0.265 and 0.268. The percentile’s errors at those levels are 0.261, 0.266, 0.269, 0.270 and 0.277. So reading an actual tail is slightly worse than a normal point times the spread, and that is noise in 1,000 resamples. On a normal law, all of a quantile’s information is its scale.

The variance read as a scale is the remaining dashed line, and it wants a little more: 7.0, 10.9 and 21.6 against the point’s 6.6, 9.8 and 18.2. That gap is real and it does grow with persistence. The earlier essay’s headline gap survives too: on the same draws its per-sample oracle read 18.92 against 16.05, and centred it reads 18.92 against 16.64 on the rectangle and 21.82 against 20.30 on the taper. So a point does want a slightly shorter block than the variance, by about one or two blocks at this persistence. What does not survive is the reason given for it. If the cause were the tail, the gap would widen as the level moves out, and it does not. The variance there is the implied closed-form sum and the point is read from a finite set of resamples, so the two are not quite one reading of one thing, and at 0.9 the error curve is flat enough around its minimum that the gap in block length is worth under a percentage point of error. The cause is not settled here.

Where a shape does show #

A Gaussian series is the kindest case for that argument, so it has to be run where a shape exists. With exponential shocks the block means are skewed, the upper and lower points are not mirror images, and a sum of few blocks is genuinely further from its limit.

There the original intuition does show up, weakly. At a persistence of 0.7, the upper point wants 9.7, 9.8, 9.3, 9.0 and 8.9 as the level rises, and the lower wants 9.4, 9.6, 9.7, 9.9 and 10.0. So on the long side of a skewed law a further point does want a slightly shorter block, by about one block between the 75% and the 99%, and on the short side it goes the other way. A real effect, of the expected sign, and a tenth of what the earlier essay’s text implied. It is the only place in this field so far where a reading depends on anything but a scale.

The two points also cost different amounts. At the 99% level the upper point has an error of 0.307 and the lower one of 0.277: the long tail is dearer to read than the short, on the same resamples.

The taper’s real advantage #

That leaves the most striking claim of the three. It said the tapered window loses to the rectangle on the variance at two rules, and beats it at all four rules on the point and on coverage: a reversal, and an argument that an ordering by an implied variance cannot be trusted.

The figure at the head of this essay is all five readings, each as the taper’s difference from the rectangle. Read the first column: the variance favours the rectangle at the protocol length and the rule of thumb, and the taper at the other two. Read the second and fourth: those are the uncentred point and the uncentred coverage, the earlier essay’s reversal, and they favour the taper everywhere. Now read the third and fifth, the same two readings about their own mean. They agree with the first column at every rule that is not a tie. There is no reversal. All three instruments order the windows alike.

Centring the point explains the third column, but it cannot explain the fifth, because coverage is a percentile interval and it was never scored against a truth with no sample in it. Something else was helping the taper’s interval to cover. It is this.

A tapered block multiplies each value by a weight. The weights have a mean square of one, so the block’s variance is right. Their mean is less than one, so its level is not: a resampled mean is the sample’s mean multiplied by the mean weight. For the usual trapezoid that is a fixed factor, computable in a line, of 0.875 at a block of twelve and between 0.873 and 0.875 at every length from eight to forty-eight. Counted over 600 samples, the resampled mean’s slope on the sample’s mean is 0.888 at a mean of zero and 0.875 once the mean is clear of zero, where it can be measured properly. A rectangle’s is 1.

So a taper’s percentile interval is not centred on the sample’s mean. It is centred about an eighth of the way back towards zero.

And every series in every essay on this line has a true mean of exactly zero. An interval that drifts towards zero is drifting towards the answer. That is a free gain in coverage that has nothing to do with how the window handles dependence, and nothing to do with any real series, whose mean is not where the construction drifts to.

The check is direct, and it is the one that any real use of the interval faces. Move the series’ true mean off zero. The rectangle’s uncentred coverage stays at 81–84% and its centred coverage at 84–86%. The taper’s uncentred coverage is 88.3% at a mean of zero, 87.0% at 0.5, 83.5% at one, and 61.7% at two spreads. Its centred form stays at 83.8–85.8% all the way along. Everything the taper had over the rectangle on coverage is gone once it is centred: 85.8% against 85.5% at a block of twelve.

Note that the percentile interval is usually written for a residual with a known zero mean, and a practitioner who runs the taper on data with a non-zero mean would have to subtract the mean first and add it back. That is the centred form. So the coverage the earlier table printed was not achievable on any data for which a trial wanted to estimate a mean.

What survives #

Quite a lot of the surrounding field does, and the correction is narrow.

The instruments’ truths are right. The finite-sample 95% point of 3.889, and its gap from the asymptotic 3.9155, are correct. Only the bootstrap’s reading was off.

What the interval covers still stands about the shortfall. Coverage of about 81% to 91% at a nominal 95% is a real failure, and it reads about 81.5% to 88.0% when centred. The rectangle’s and the taper’s relative order there has moved, so that essay’s reading of a rule “inverting” between windows needs the same correction. Its fourteen-point shortfall has shrunk by nothing.

The field’s cross-check on the variance is unaffected. It used scale differences at a fixed block, and centring does not change a width.

The recommendation against a block length of 4 stands, and it now has all five readings behind it.

What does not survive: that the ordering of the windows depends on the instrument; that the ordering by a variance is untrustworthy on that account; that a quantile is dearer; that a quantile’s shorter block comes from its tail; and that a tapered block has a coverage advantage over a rectangle at these lengths. What is left of the case for a taper is the case the variance always made: at a good rule, it has a modest edge, 1.1 to 2.1 points of relative error, and at a bad rule a modest loss, for reasons the implied autocovariance gives directly.

The earlier essays keep their text and their figures, since they are accurate records of what they computed, and each now carries a note at its head pointing here. Their tables are uncentred and should be read as such.

How it was missed, and what to carry from it #

Nothing failed. The test on the bootstrap asks whether its readings come out of one sorted array in the right order; a uniform shift keeps the order. The comparison with a closed form was done for the variance, where the identity holds with or without a mean, and not for the point, because there is no closed form for it. And the numbers looked reasonable: a quantile being dearer than a variance is what anyone would expect, on the argument that a tail is harder to estimate. A result that agrees with an expectation does not get audited.

The tell, in the end, was a level nobody needed. At a 75% point the truth is small and the sample’s mean is not, so the error came out at 162%, which is not a plausible figure for a bootstrap of a Gaussian mean. A sweep over the level was enough to show it: a reading’s relative error should not depend on how close the truth happens to be to zero, unless the reading has something else in it. The same check would have found the taper’s shrink sooner if any series on this line had carried a mean. All of them were generated as residuals, for a good reason, and that reason is exactly what hid this.

And a measurement that compares two arms for a thing that moves with the truth should try moving the truth. That is the whole of the mean sweep, it took one parameter, and it is the one that shows the effect directly.

How the numbers were made #

The rule comparison re-runs the earlier essay’s sweep on its own seeds, 400 draws with 300 resamples at seven block lengths, so its uncentred figures reproduce that essay’s to the digit: 45.83%, 65.51%, 84.8% and the rest. Centring subtracts each resample set’s own mean from the point, and for coverage adds $\sqrt{n}\,\bar{x}$ back, on the same resamples. The level sweep is 300 draws with 1,000 resamples at each of seven lengths, with each level’s truth from 40,000 series. Best lengths are read off a parabola on a log-length axis through the three lowest points. The skewed law is a first-order autoregression whose shocks are a centred exponential, scaled so its marginal variance is 1.

Five things must hold. The centred point must be cheaper than the variance at all four rules. The centred point and the variance must order the windows alike at all four, while the uncentred point on the same draws shows the earlier essay’s two flips. A rectangle must leave the mean alone, and a trapezoid must shrink it by a factor below 0.92. With the mean moved to two spreads the uncentred taper must fall below 75% and the centred one stay above 80%. And across the five levels, at each of three persistences, the largest centred best length must be within 15% of the smallest. That last one refuses the essay this one set out to write.

Still open: the other essays this touches #

The audit is not finished. Six other essays quote the instrument essays’ figures, and the field on the studentised interval picks its oracle block with the same uncentred point. That studentised interval is centred by construction, so its coverage is not affected, but its oracle row and its percentile comparison column may move. What studentising costs and the ordering that reverses again are the first to reread. This essay names them and does not correct them.

A window with a level. A taper with a mean weight of one instead of a mean square of one would keep the level and shrink the variance, and that is the same trade pointed the other way. Whether there is a family of weights that does both, by an offset rather than a scale, and whether it is any different from centring, is a short question. It may matter for a resampled statistic that is not a mean, where there is nothing to subtract.

What this retracts and what it still claims #

This essay takes the bootstrap reading of a mean’s 95% point, and the coverage of its interval, on a tapered and a rectangular window, at a persistence of 0.7 on 120 rows. Its claims are that read about its own mean the point costs 28.48% against the variance’s 45.83% at a protocol length, and is the cheapest reading at every rule; that across five levels and three persistences the block a Gaussian point wants is flat; that a skewed law moves the block by about one between the 75% and the 99% point, in opposite directions on the two tails; that a trapezoidal taper shrinks a resampled mean by 0.875 in closed form; that its uncentred coverage falls from 88.3% to 61.7% as the true mean moves from zero to two spreads, while its centred coverage stays near the rectangle’s; and that centred, the variance, the point and the coverage order the two windows alike at all four rules.

It retracts the first two claims named at the top, and the third’s mechanism: the small gap in block length stands, and the reason offered for it does not. What stays out: other statistics than a mean, where centring has a different meaning; the audit of the other quoting essays, recorded above; and a persistence other than 0.7 for the ordering, since the earlier sweep was run at one. The boundary against the instrument and the reading is that it asked what three readings of one bootstrap say about one comparison, and this one is the same question, asked of a reading that had been centred.