A ranking that belongs to the interval

A block bootstrap for the mean of a dependent series resamples blocks of consecutive rows, and the block length is a tuning parameter nobody is given. This field has priced the rules a practitioner could use to choose it: a length written into the protocol, the rule of thumb $n^{1/3}$, the plug-in that reads the sample’s lag-one correlation and assumes the rest of the dependence is geometric, and — in a length read off the whole sequence — a rule that reads every autocovariance through a flat-top window and so assumes nothing about the shape. Beside them is the oracle, the length that would have been best on the draw, which needs the truth.

Every one of those comparisons was made on the implied variance: the long-run variance a block length’s resampling implies, computed exactly rather than resampled, and scored against the variance of the mean it is aimed at. That economy is what made four sample sizes and three laws affordable. The last essay found the whole-sequence rule buying nothing at 120 rows under a five-period moving average — 41.5% error against the plug-in’s 43.4%, with a fixed eight lags at 35.4% better than both — and separating at 480. And it named the obvious objection to all of it. A practitioner never reads a long-run variance. They read an interval. The instruments field had already shown that a ranking can reverse between the variance and the interval it builds, and the ordering reverses again had shown it reversing once more between two intervals built from the same resamples. This essay reads the five rules on the interval.

Four intervals from one set of resamples #

Each draw is a series of 120 or 480 rows from one of three laws — a first-order autoregression at 0.7, a five-period moving average with nothing past the fourth lag, and long memory at a fractional difference of 4/9 — with a true mean of zero. Each rule picks its length from the sample, the oracle picks the length whose implied variance is nearest the truth on that draw, and at each picked length four hundred moving-block resamples are drawn. From those same resamples four intervals are built.

The percentile interval reads the 2.5th and 97.5th points of the resampled means; it is the interval every earlier essay in the line counted. The normal interval is the sample mean plus or minus 1.96 resampled standard deviations. The studentised interval divides each resampled mean by its own standard error from its own blocks and reads its critical values from that, the block bootstrap-$t$. And the batch-$t$ interval resamples nothing: it cuts the sample into whole blocks of the chosen length, takes the standard error of their means, and multiplies by Student’s $t$ on one fewer degree of freedom than there are blocks — the construction the interval with no resampling in it found covering at least as often as the studentised one under the first-order law. Two thousand draws at each of six settings, and every comparison below is paired on the draw.

The picture’s shape is the essay’s first finding. On each row the five rules sit close together, and between the two rows of a setting the whole cluster moves. Under the moving average at 120 rows the five percentile coverages run from 85.3% to 88.8%, a spread of three and a half points; the same five on the studentised interval run from 87.9% to 95.0%, and the plug-in alone moves from 86.4% to 94.9%. Under the autoregression at 120 rows the plug-in moves from 85.8% to 92.0%. The choice of interval moves a rule’s coverage by up to eight and a half points, and the choice among the four feasible rules moves it by at most a few. That is the same ordering of dials the instruments field reported for the window, and it holds for the length rule too: what is built from the length matters more than how the length was chosen.

The rule of thumb is the exception that proves the point. It is the one rule that picks short blocks — four lags at 120 rows — and it is last on every interval at every short-sample setting, because a block of four cannot carry a dependence that runs past it, and no interval can recover what the resampling never saw.

The variance’s verdict, read on the interval #

The lead asked a specific question: whether the variance’s ranking of the two estimated rules, the plug-in and the whole-sequence rule, survives on the interval a practitioner acts on.

On the percentile interval it mostly does. At 120 rows the variance had the two estimated rules level, 43.4% and 41.5% of error, and the percentile interval agrees: 86.4% and 86.3%. The variance had the protocol’s eight lags ahead of both, and the interval agrees again, at 88.5%. At 480 rows the variance put the whole-sequence rule ahead of the plug-in, 26.1% of error against 30.0%, and the interval puts it ahead too, 91.6% against 90.4%. So the plug-in’s one assumption — that the dependence is geometric — costs it on the interval what it cost it on the variance, at the sample size where it costs anything.

There is one comparison the two readings disagree about, and it is the protocol’s length at 480 rows. On the variance the fixed eight lags is ahead of the sequence rule, 25.1% against 26.1%; on the percentile interval it is behind, 90.5% against 91.6%. The variance scores the size of an error and is indifferent to its sign; a short block’s implied variance errs low, and a percentile interval built on a low variance is short, which coverage punishes and a squared error does not distinguish from erring high. That asymmetry is the whole reason a ranking can move between the two instruments, and here it moves one pair out of ten.

And reversed by a second interval #

The percentile interval is not the only interval, and on the studentised one the same comparison comes out the other way.

Under the moving average at 480 rows the whole-sequence rule leads the plug-in by 1.2 points on the percentile interval and trails it by 1.1 points on the studentised one, 93.3% against 94.5%, each difference about two and a half paired standard errors from zero. At 120 rows the percentile interval has them level and the studentised one has the plug-in ahead by two points, 94.9% against 92.6%. Under the autoregression the plug-in is ahead on the studentised interval at 120 rows and the two are level at 480. Every one of the reversals runs the same way: on the intervals that widen for their own uncertainty — studentised and batch-$t$ — the plug-in’s longer blocks win.

The mechanism is the plug-in’s defect turned into a virtue. Under the moving average its assumed geometric dependence reads the lag-one correlation of 0.8 as a long memory, and it picks blocks about twice as long as the law needs — 18 lags at 120 rows, 30 at 480, against the sequence rule’s 12 and 16. On the variance that is an error, and on the percentile interval it costs, because a long block from a short series is resampled with too little variety, its resampled spread falls, and the percentile interval shrinks with it. A studentised interval does not shrink: fewer blocks means a noisier standard error, and the resampled $t$ statistic’s tails grow to match, the same way Student’s $t$ on fewer degrees of freedom does. So a long block costs a percentile interval coverage and buys a studentised one coverage, and which estimated length is “better” depends on which of the two intervals it is for.

Which half of the interval does it #

The four constructions were chosen so that this question has an answer. The percentile and normal intervals both take their scale from the spread of the resampled means and their critical value from the normal law, in effect; they differ only in whether the resampled distribution’s shape is kept. The studentised and batch-$t$ intervals both take their scale from the blocks’ own standard error and a critical value that grows as the blocks get fewer; they differ only in whether anything is resampled. If the reversal came from the resampling — from the shape of the bootstrap distribution, or its skewness — the normal interval would side with the studentised one, or the batch-$t$ with the percentile one. Neither does.

Under the moving average at 120 rows the whole-sequence rule’s paired lead over the plug-in is −0.10 points on the percentile interval and −0.55 on the normal one, both within a standard error and a half of nothing; it is −2.30 on the studentised interval and −2.20 on the batch-$t$, both more than four standard errors from it. At 480 rows the four are +1.20, +0.20, −1.10 and −0.30: the split is weaker there and the studentised interval carries most of it. The two intervals with a normal critical value side with each other, the two with a critical value that counts its blocks side with each other, and the split between the pairs is the reversal. What decides which length rule wins is not how the bootstrap distribution is read but whether the interval’s multiplier knows how many blocks its standard error rests on — the property what studentising costs found making the studentised interval twice as wide at the lengths the rules choose, and the count or the length traced to the block count alone.

That also explains why the protocol’s eight lags behaves as it does. At 120 rows eight lags leaves fifteen blocks, enough that Student’s $t$ on fourteen degrees of freedom is close to the normal, so its studentised and percentile intervals are close; the plug-in’s eighteen lags leaves six blocks, and its studentised interval is a different object from its percentile one. The rules that pick long blocks are the rules whose ranking the interval decides.

That is the ordering reverses again in a new place. There the reversal was between two block windows read on two intervals; here it is between two length rules, under a law one of them was not written for. The general statement the two together support is that a block length is not good or bad; it is good or bad for an interval, and a rule recommended on one interval’s coverage is not thereby recommended on another’s.

Long memory, where nothing covers #

The third law is a different kind of answer.

The last essay found every rule’s implied variance short of the mean’s true variance by more than ninety per cent under long memory, including the oracle’s. Read as an interval, that is coverage between 14.3% and 26.9% for the percentile interval at every rule and both sizes. The studentised interval does better where the block is longest — the oracle’s length covers 60.8% at 120 rows, where it picks blocks of about twenty-nine lags and leaves four of them — and the sequence rule, which under long memory picks the longest feasible blocks, reaches 35.7% at 120 rows and 34.4% at 480. Nothing reaches two in three. And a longer sample makes every short block worse: the protocol’s eight lags covers 23.1% on the percentile interval at 120 rows and 14.3% at 480, because the dependence the block truncates is a larger share of the variance the longer the series runs.

That is not a ranking at all. Under long memory the variance of the mean does not shrink at the rate a block bootstrap assumes, no block length short of the sample carries the dependence, and the interval’s failure is the method’s rather than the rule’s. The choice among rules there is a choice among intervals that each miss more often than they cover, and the honest reading is the one the previous essay gave in variance terms: a block bootstrap is the wrong instrument for a long-memory mean.

What the coverage costs #

The studentised interval’s advantage is not free. Under the moving average at 480 rows its median width at the plug-in’s length is 0.411 against the percentile interval’s 0.360, a seventh wider, for four points of coverage. The batch-$t$ interval is the same width to the third decimal and covers as well, with no resampling at all. On those two constructions coverage rises with width, and the rule that picks the longest block covers best. On the percentile and normal constructions it does not: the plug-in’s blocks, the longest any feasible rule picks, give a percentile interval that covers no better than the protocol’s eight lags at a wider width. A reader choosing a length rule for a percentile interval and one choosing for a studentised interval should choose differently, and the figure is the reason.

How the numbers were made #

Two thousand draws at each of six settings, each series generated exactly from its law — the autoregression by its recursion, the moving average and long memory through a dense Cholesky factor of their autocovariance matrices. Each rule reads only the sample: the protocol’s eight lags, $n^{1/3}$ rounded to the grid, the plug-in from the lag-one correlation, the whole-sequence rule from the flat-top-windowed autocovariances with Politis’s cut-off. Every length is snapped to the field’s grid of fourteen and restricted to lengths that leave at least three whole blocks. Four hundred moving-block resamples are drawn once at each picked length on each draw, using whole blocks so every resample has blocks to studentise with, and the four intervals are read from them; the batch-$t$ interval uses non-overlapping blocks of the same length. Coverage is of the true mean, zero.

What is checked on the measurement: the studentised interval covers more than the percentile one at every rule’s length at all four short-memory settings; under the moving average at 480 rows the whole-sequence rule leads the plug-in on the percentile interval and trails it on the studentised one; and under long memory no rule and no interval covers two times in three.

Still open: a length chosen for the interval it builds #

The finding points at a rule nobody has written down: a block length chosen for the interval it will build, not for the variance it implies. For a studentised interval the trade is between the bias a short block leaves and the width a long one costs through a small number of blocks, and both halves have closed forms in this field — the implied variance’s bias, and Student’s $t$ on the block count. A rule minimising expected width subject to a coverage floor would pick different lengths from every rule here, longer under the moving average and the autoregression, and it would have to be scored on draws it was not tuned on. The prediction from the figures is that it lands near the oracle on the studentised interval and nowhere near it on the percentile one. It has not been built.

What is claimed here, and what is not #

This essay takes the field’s five block-length rules read on the coverage of four intervals for the mean. The claims are that across the short-memory settings the choice of interval moves a rule’s coverage by up to eight and a half points and the choice among feasible rules by a few; that under the moving average the percentile interval agrees with the implied variance that the plug-in and the whole-sequence rule tie at 120 rows and that the sequence rule leads at 480, and disagrees with it about the protocol’s eight lags at 480; that on the studentised and batch-$t$ intervals the plug-in’s longer blocks lead instead, by about one point at 480 rows and two at 120 under the moving average; that under long memory no rule and no interval covers two times in three; and that the studentised interval’s extra coverage costs about a seventh in width at the plug-in’s length.

What stays out, named as a decision: other windows, since every resample here is the flat moving-block one; other statistics than the mean; and a length rule built for the interval, deferred above.

The boundary against a length read off the whole sequence is that it compared the rules on the implied variance and this essay reads the same rules on the coverage of what they build. Against the interval with no resampling in it and the field it belongs to, which compared intervals under the first-order law with lengths from a fixed grid and the three older rules, this essay holds the intervals fixed and asks which length rule each one prefers, under two laws that field did not draw.