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

Bootstrap — the series

24 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Where the bootstrap works and where it does not. Uniform data on [0, 1]. For the mean the percentile bootstrap covers 93.5%. For the maximum it covers 0.0%, because a resample can never contain a value larger than the largest one observed, so the interval cannot reach above it.

    Where the bootstrap lies

    Resampling is the most generally useful trick in the subject and it has a failure mode that is easy to state: it cannot see past the data. For a statistic that lives at the edge of the sample, coverage collapses from 95% to almost nothing.

    part 3 · intervals
  2. The distribution the table does not have. 599 series simulated from the smaller model fitted to one comparison's own data, the whole rolling comparison re-run on each, and the ordinary statistic recorded. Under this null the two forecasts are the same forecast in population, so what is left in a sample is the larger model's estimation error and the statistic is centred at -1.134 rather than at zero. Its 95% point is 0.264; the standard normal drawn behind it puts that point at 1.645. Reading this statistic against that curve is not a poor approximation, it is a different distribution: the share of this one above 1.645 is 0.2%.

    A distribution drawn from the null

    Between nested models the ordinary comparison statistic has a null distribution centred at minus one and a 95% point of a quarter. A correction to its mean repairs the centre and leaves the shape; simulating the null repairs both.

    part 4 · ranking
  3. The distribution of the largest statistic in the table. Fit the benchmark to the whole series, resample its residuals, simulate 199 series in which the null is true by construction, re-run the entire eight-variant search on each, and keep the largest statistic. That is the distribution drawn here, and it is the distribution of the thing a specification search actually reports. It is centred at 1.045 — the maximum of eight statistics is not centred at zero however well each of them behaves — and its 5% point is 2.536. A table read against 1.671 is reading the distribution of one statistic; a Bonferroni correction reads it against 2.577 and is nearly right here, because eight variants that each add a different lag are nearly eight separate chances.

    A null with a model in it

    The distribution to read the winner of a table against cannot be resampled from the data, because the data does not contain the null. It has to be generated from a model — which is the assumption the resampling was chosen to avoid.

    part 5 · search
  4. Which repair goes with which defect. The share of true nulls rejected at a nominal 5% by a reference distribution generated from the fitted benchmark, over 120 draws with 59 resamples each. Where the errors are well behaved every resampling is fine and all four are conservative. Where the variance is a function of the design, the two that detach a residual from its own row reject 5.8% and 8.3% — and the block bootstrap, which is the resampling three earlier fields on this site reach for, repairs nothing at all, because the dependence it is built for is between origins and the rolling scheme reproduces that on its own. Where the errors are skewed the symmetric multiplier is the one that is wrong, and Mammen's two-point version is the only one of the four that is right in both columns.

    Residuals that keep their own variance

    A reference distribution for a search has to be generated from a fitted model, and the generator draws residuals. Four ways of drawing them keep four different things — and the one this site has reached for three times repairs nothing at all here.

    part 6 · select
  5. Each repair is for its own defect, and one is for both. The 95% point of the statistic's own distribution in each world, against the mean 95% point of five reference distributions built from one sample. Where the error variance is a function of the design, the two resamplings that detach a residual from its row fall short and the two multipliers that keep it there do not; where the rows repeat each other it is the other way round. With both defects at once the blocked multiplier — drawn once per run of 5 rows, so the residual never moves and its neighbours share a sign — is the closest of the five, at 2.999 against a truth of 3.803. It is still short by 0.804, and that shortfall is the next figure.

    Two defects and one resampling

    Four resamplings, each the repair for one defect and wrong about the other. Put both defects in the same world and the statistic's 5% point is 3.8028, where the best of the four reaches 2.8326 — until a multiplier that stays on its own row and shares a sign with its neighbours reaches 2.9988.

    part 7 · proxy
  6. A fit takes the low frequencies out of what it leaves behind. The autocorrelation of the errors, of the residuals of a fitted benchmark, and of those residuals rescaled by their own leverage. (I − H) removes the component of the errors lying in a column space that is itself slow-moving, so the residuals are less persistent at every lag — by 5.9% at the first and 26.6% by the fourth. The leverage correction is the standard repair for what a fit does to a residual's size; drawn here against what it does to a residual's dependence, it does nothing.

    The residuals are not the errors

    A fit removes the part of the errors lying in its own column space, and a persistent design's column space is itself slow — so what is left behind is smoother than what went in, at every lag, by an amount that grows with the lag.

    part 8 · effective
  7. Two constructions on one triangle, and a third that is not. Three resamplings that all keep runs of neighbours, on the same residuals at a block length of 5, with the lags running past ℓ so that the tapers separate. A blocked multiplier never moves a residual; a fixed-length moving block moves every one; and they attenuate identically, worst gap 1.4 standard errors, both sitting on γ_resid(k)(1 − k/ℓ)⁺ and both exactly zero past ℓ — so the attenuation is the block boundary rather than the multiplier. The third is the stationary bootstrap, whose runs are geometric rather than fixed: its taper is γ_resid(k)(1 − 1/ℓ)^k, it agrees with the other two at the first lag and at no other, and at lag 6 it still carries 0.0081 where they carry -0.0005.

    The triangle that was not the multiplier's

    A resampling that leaves each residual on its own row can keep only what the residuals have, times a triangle. A construction that moves every one of them has the same triangle — and the one in this collection's own table has a different taper entirely.

    part 9 · banded
  8. 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.

    part 10 · taper
  9. What a sample shows, and what the algebra does. The difference between a rectangular block's implied long-run variance and a trapezoidal one's, as a share of the truth. Above the axis the rectangle is less biased and below it the trapezoid is. The heavy line is exact — computed from the law's own autocovariances — and it crosses at 19.2. The others are what samples of 120, 240, 480, 960 rows report, and every one of them exaggerates whichever window is ahead: at ℓ = 20, where the exact difference is 0.28 points, a sample of 120 rows shows 4.31 points — 15 times larger. That is the number the earlier reading of this comparison was missing: three tenths of a point is what the algebra says and not what a hundred and twenty rows report.

    The gap a sample shows

    The exact difference between two block windows at a block length of twenty is three tenths of a point. What a hundred and twenty rows report is four and a third, because the autocovariances the window is applied to are attenuated too.

    part 11 · crossing
  10. What a longer block buys and what it costs. A trapezoidal block at 120 rows, with the error split into the two things it is made of. The bias falls with the block length, because a longer block attenuates less, and it flattens at 23.2% because the sample's own autocovariances are short whatever window is applied to them. The spread rises with it, because a longer block means fewer of them. Their sum in quadrature has a minimum at ℓ = 16, which is not where either of the two has one. The faint line is the rectangle's total error, for scale: it is above the trapezoid's from ℓ = 12 onwards.

    Bias is not the whole of it

    A window that reaches zero at its ends attenuates less and uses less of each block. The block length that minimises its bias is not the one that minimises its error, and comparing two windows at one length compares one of them mis-tuned.

    part 12 · crossing
  11. What each instrument costs to read. The number of draws each instrument needs to separate a rectangular block from a trapezoidal one at two standard errors, at a block length of 20 and 120 rows — measured from each instrument's own spread on the same draws. The implied variance needs 7.0 and the 95% point needs 20.2, a factor of 2.90 at this block length. There is a closed form beside it and it does not depend on either the scale or the size of the gap: the standard error of a p-quantile is √(p(1−p))/f(q) over √B where a standard deviation's is σ/√(2B), which at the 95% point of a nearly normal reference distribution is 3.30 times as many draws for the same statement. And the quantile route needs every one of those draws resampled, where the variance route needs none.

    Measuring a variance rather than a quantile

    A resample's implied long-run variance can be computed from the sample with no resampling in it at all. A critical value cannot, and the difference is a factor of three in the draws before any of the resampling is counted.

    part 13 · crossing
  12. Where the taper's case begins, and it is not where the algebra says. The block length at which a trapezoidal block's implied variance stops being more biased than a rectangular one's, against the length of the sample. Computed exactly — from the law's own autocovariances, with no sampling in it — the answer is 19.2 and does not depend on the sample at all. What a sample of 120 rows reports is 13.3, and the reported crossing walks out towards the exact one as the sample grows: 13.3, 15.0, 16.4, 18.0. The mechanism is that the autocovariances the window is applied to are themselves attenuated, worst at the longest lags, and the window that discards those lags loses less of them.

    The error no window repairs

    Every block window's best estimate of a long-run variance is wrong by about forty per cent at a hundred and twenty rows, and the largest part of that is not a bias at all. Choosing the window moves a twentieth of it.

    part 14 · crossing
  13. 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.

    part 15 · feasible
  14. 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.

    part 16 · readout
  15. 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.

    part 16 · feasible
  16. 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.

    part 17 · readout
  17. 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.

    part 17 · feasible
  18. 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.

    part 18 · readout
  19. 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.

    part 19 · readout
  20. Three intervals, one shortfall. What each of three intervals actually covers, at four rules and two block windows, over 300 samples of 120 rows. All three are built from the same resamples on the same draws, so a difference between them is a difference in what is done with the resampled series. Not one of the twenty-four cells reaches the ninety-five per cent it promises. The studentised interval runs from 75.7% to 92.3%, the percentile interval — the earlier field's — from 80.0% to 89.7%, and a normal interval on the same scale from 81.7% to 89.0%. The standard repair for a percentile interval's shortfall does not repair it.

    An interval that carries its scale

    A percentile interval inherits the resampled distribution's skewness and its scale error together. The standard repair is one extra variance per resample. It was named and not run, so this runs it.

    part 20 · student
  21. What studentising costs. How much wider the studentised interval is than the percentile one, cell by cell, over 300 draws, with what each cell gains in coverage beside it. Averaged over the eight cells the interval is 2.09 times as wide and covers 0.46 points better. At the two rules that choose short blocks the two intervals are within a fifth of each other; at the oracle's length, where a resample holds two or three whole blocks, the studentised interval is 4.37 and 5.04 times as wide. A repair that doubles the width and buys half a point is not one a reader could not have had by widening the interval it replaced.

    What studentising costs

    Averaged over eight cells the studentised interval is 2.09 times as wide as the percentile one and covers 0.46 points better. At the block lengths the rules choose, the scale it divides by rests on two or three numbers.

    part 21 · student
  22. Four rules of four change sign. The margin between the two block windows in points of coverage, under each of four rules, on each of three intervals built from the same resamples, over 300 draws. Positive is the tapered window covering better. On the percentile interval the taper wins at all four rules, by 5.33, 1.67, 5.00 and 4.00 points, which is the earlier field's own reading. On the studentised interval the rectangle wins at all four, by 4.33, 7.00, 4.33 and 2.67. And a normal interval, which uses no resampling at all, puts the two within a third of a point at every rule — so the disagreement is manufactured entirely by what is done with the resamples.

    The ordering reverses again

    One field found two of four rules changing sign between two readings of one resampling. Turn the same resamples into a studentised interval instead of a percentile one and all four change sign.

    part 22 · student
  23. One penalty, read along two dials. How much wider the studentised interval is than the percentile one, at every sample size and every block length on the grid, with the number of whole blocks each cell leaves written beneath. Read across a row and the block length changes; read down a column and the sample size does. The penalty is nearly a function of the block count alone: the cells at 15 blocks read 1.16, 1.20, 1.17, 1.15, 1.13, 1.10 across three sample sizes and three block lengths, while the cells at one block length read anything from 1.10 to 3.95. The largest penalty on the grid is 3.95, at the cell with 3 whole blocks in it.

    The count or the length

    A block length and a block count are one number read two ways at one sample size. Read at three, the studentised interval's width penalty tracks the count — with an R² of 0.9911 against a closed form that has no length in it — and its coverage tracks the length.

    part 23 · student
  24. Four intervals at 3 blocks of 32 rows. What four 95% intervals for the mean of a first-order autoregression at 0.7 cover, and how wide they are on average, at 120 rows cut into 3 whole blocks of 32, over 240 draws with 200 resamples each under the rectangle. The normal interval, the block-means variance with 1.96, covers 82.1% at a width of 0.708. The percentile interval covers 82.1% at 0.632 and the studentised one 90.4% at 2.495. The fourth resamples nothing: it is the normal interval with 1.96 replaced by Student's t on 2 degrees of freedom, and it covers 94.2% at 1.555, 0.62 times the studentised interval's width.

    The interval with no resampling in it

    Replace 1.96 in a normal interval on the block-means variance with Student's t on one fewer degrees of freedom than there are whole blocks, and resample nothing. Across twenty-four cells it covers at least as often as the studentised bootstrap interval at every one, by 0.42 to 10.42 points; it is narrower wherever seven blocks or fewer are left; and at fifteen blocks of 32 it covers 95.0%, which no resampled interval on the grid reaches.

    part 24 · student

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