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

A block length chosen from the data

Two block windows were compared at each window's own best block length, which is the argmin of a quantity that needs the truth. Re-run on rules a practitioner could actually run, the ordering reverses: the taper wins at the best available length and at one estimated from the sample's own persistence, and the rectangle wins at a length written into a protocol and at the rule of thumb, at every sample size measured. And the whole argument is a third of the size of what estimating the length costs.
Three rules and a target none of them is aimed at. Which block length each rule picks, over 400 samples of 120 rows, for the tapered window. Two of the rules are points: a length written into a protocol is 8.00 on every draw and the rule of thumb is 4.00, because n to the one third does not read the data at all. The plug-in reads the sample's own persistence and lands at 14.36 with a standard deviation of 2.93. The length that would actually have been best on that draw averages 24.57 with a standard deviation of 16.23 and runs from 10 to 48 between its tenth and ninetieth percentiles. The target moves five times as much as the best estimate of it does, which is why no rule can be close to it and why the two that do not try are not merely worse — they are somewhere else.

The length nobody has

Every comparison of block windows in this collection is made at each window's own best block length. That length has a standard deviation of sixteen across draws and averages twenty-five. No rule is aimed at it.

 Which window is better depends on who chose the block length. The margin between a rectangular block and a tapered one, on 400 samples of 120 rows, under four rules for choosing the block length. At the length that would actually have been best on each draw the taper is ahead by 2.12 points of a 35.8% error, at 22.1 paired standard errors; at a length estimated from the sample's own persistence it is ahead by 1.89. At the length this field's own figures use — eight — the rectangle is ahead by 2.04, and at the rule of thumb by 4.54. Every rule sees the same draws. What separates them is the length: the two rules that lose to the rectangle pick 4.00 and 8.00 where the best available is 24.57, and a tapered window at a quarter of the right length has thrown away most of what it was weighting.

An ordering that depends on the rule

The tapered block beats the rectangular one at the best available block length and at one estimated from the data. At a length written into a protocol, and at the rule of thumb, the rectangle wins — at every sample size measured.

 The argument is a third of the size of the thing it is inside. Three quantities on one scale, in points of the error in a block resample's implied long-run variance, at 120 rows. The gap between the two windows at the best available block length — the whole subject of the comparison this field inherited — is 2.12 points. What the best rule a practitioner could actually run gives up against that same best length is 7.26, a factor of 3.42. What the rule of thumb gives up is 26.01. So the ordering between windows is worth establishing and is not worth arguing about, and the sentence that follows from it is not use the taper but estimate the block length, because that is where the points are.

What choosing the length costs

The gap between two block windows at the best available length is 2.12 points. What the best rule a practitioner could run gives up against that same length is 7.26. The argument is a third of the size of the thing it is inside.

 A premium that shrinks and a payout that grows. The relative error of the trapezoid's implied variance with the block length read off the whole autocovariance sequence, minus its error with the length from the lag-one correlation alone, in points, at four sample sizes and 400 draws each. Under the first-order autoregression the plug-in's algebra was written for, reading the sequence costs 3.87 at 120, 2.37 at 240, 2.48 at 480, 1.15 at 960. Under the five-period moving average, which the plug-in reads as an autoregression at 0.8, it is 0.43 at 120, -1.72 at 240, -3.01 at 480, -3.77 at 960 — a tie at 120 rows and a gain that grows with the sample.

A length read off the whole sequence

The automatic block length reads one number off the sample — the lag-one correlation — and assumes the rest of the dependence is geometric. Reading the whole autocovariance sequence instead removes the assumption and puts the length where the law wants it. At a hundred and twenty rows it buys nothing, even where the assumption is wrong; it costs 3.87 points where the assumption is right, and by 960 rows that premium is 1.15 and the payout under a law the plug-in misreads is 3.77. Under long memory neither rule, nor the best length on each draw, gets within ninety per cent.

 Five length rules, read on two intervals. The coverage of a 95% interval for the mean at the block length each of five rules picks, on the percentile interval and the studentised one built from the same resamples, under a five-period moving average and a first-order autoregression at 120 and 480 rows, 2,000 draws each. moving average at 120: percentile 88.5%, 85.3%, 86.4%, 86.3%, 88.8%; studentised 92.5%, 87.9%, 94.9%, 92.6%, 95.0%. moving average at 480: percentile 90.5%, 90.5%, 90.4%, 91.6%, 91.9%; studentised 91.2%, 91.2%, 94.5%, 93.3%, 95.0%. autoregression at 120: percentile 84.7%, 80.2%, 85.8%, 85.4%, 85.9%; studentised 90.3%, 83.0%, 92.0%, 90.9%, 94.3%. autoregression at 480: percentile 87.0%, 87.0%, 90.0%, 90.5%, 90.9%; studentised 88.6%, 88.6%, 93.0%, 93.0%, 94.2% (in the order eight, from the protocol, n to the one third, the plug-in, the whole sequence, the best on the draw). The rules spread over at most 5.6% on the percentile interval; the interval moves the same rule by up to 8.5%.

A ranking that belongs to the interval

Five ways of choosing a block length, read on the coverage of the interval each length builds rather than on the variance it implies. On the percentile interval the variance's verdict mostly survives: under a five-period moving average the plug-in and the whole-sequence rule tie at 120 rows and the sequence rule leads at 480. Build a studentised interval from the same resamples and the lead reverses — the plug-in's longer blocks cover 94.5% against 93.3% — because an interval that widens for its own uncertainty is paid for a long block where a percentile interval is punished for one. Under long memory no length and no interval covers two times in three.

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