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

Bandwidth — where it appears

The number of lags an estimate of a dependence keeps, which decides how much of a series' memory the estimate can represent. It is chosen rather than known, and the choice is a trade: too few lags miss real correlation and too many estimate noise, so every rule for setting one lands somewhere short of the best available.

Named by 17 essays across 6 fields — each of them below, with the objects they name alongside it.

The area under the window is what the band actually costs. The three windows' weight sequences at a width of 30 lags, drawn against the lag as a share of the window. A truncated window applies a weight of one to every lag inside it and zero outside, which is why its sum is the width and why every conventional charge is right for it — and it is a covariance matrix on almost no sample, so it cannot be used. The Bartlett window falls linearly to zero and its weights sum to exactly 15.000000000000004, which is half the width, at every width: Σ(1 − k/(L+1)) over k = 1 … L is L − L/2. The Parzen window sums to 11.13 here, three eighths of the width, and it gets there by holding a weight near one over the first few lags and then falling faster. A plug-in estimate multiplied by a weight below one is a shrunk estimate, and a shrunk estimate is worth less than a free one — which is the whole of why a charge levied per lag is a charge for parameters the window has already spent.

The charge nobody derived

A band of lags is charged one log-likelihood unit apiece, because that is what a regression coefficient costs. A band's numbers are not regression coefficients, and measuring what they actually cost puts the convention out by a factor of nearly three.

dimension · Criterion The number the comparison was missing. What it costs to choose the tuning parameter for every candidate separately rather than once for the table, under AR(1) at 0.8, paired on the draw. The window's figure is the one the earlier field reported; the order's is the one it named and did not make. They are the same size — 0.00401 against 0.00360, at 2.30 and 1.72 paired standard errors — and matching the lists at eight values leaves them the same size again. The prediction that the longer list would make the order's cost the larger of the two is not what happens; what happens is that the two rules cost the same once they are scored by the same criterion, which took a missing term to arrange.

The comparison that was not made

Choosing a whitening's window separately for every candidate costs 0.00401 of regret. The same question about an order was named and left, because the two lists are different lengths. The order's answer is 0.00360, and matching the lists changes almost nothing.

lists · Order selection Two searches find some of the same luck. What each search reports on a sample with no break in it, and what the two report together, on four dependences. The dashed line is the sum of the two — what a rule charging each search separately would levy — and the two together always come in below it: 19.30, 15.44, 14.24, 26.64 short, on 100%, 99%, 99%, 100% of draws. The shortfall is not a rounding. Under AR(1) at 0.8 it is 19.30 of the 34.70 the break search manufactures on its own, which is more than half of it. Two searches over one sample are looking at the same noise, and the second one has less left to find.

Two searches, one sample

A searched break in a regression manufactures 34.7 of likelihood ratio where a count of coefficients says 11.1. A searched window manufactures 84.0. The two together manufacture 99.4, not 118.7.

twice · Break point Three charges, and only one of them is a test. What each of three thresholds does to the same decision, under AR(1) at 0.8, against the size of a genuine break in the mean at row 60. A chi-square on the 5 coefficients a split adds — 11.07 — declares a break on 73.6% of samples that have none: it is not a test at all. The break search's own 95% point, 69.6, carried into a rule that also chooses its window, fires on 0.0% of null samples and on 0.0% of samples with the largest break measured — the natural way of combining two published corrections does not lose a little power, it switches the test off. The calibrated charge, 27.2, holds 5.6% at no break and reaches 29.2% at the largest.

The charge that is not a sum

Charging two searches what each costs on its own is conservative, and conservative here means the test never fires. At the largest break measured it declares nothing, on every draw, while a calibrated threshold reaches 29%.

twice · Break point Four windows, one line, and one that is off it. The optimism measured for each window at a band of 30 lags, against what that window's weights sum to, on 2000 pairs of independent samples of 120 rows. The diagonal is where a window that spent exactly its summed weights would sit. Three of the points are one shape at three levels — the Bartlett window, its square and its cube, whose sums stand in the ratio 6 : 4 : 3 — and they lie on a line through the origin at 0.767 of the diagonal, with 0.033 between the highest and the lowest. Scaling the weights scales the charge by the factor the weights predict, which is what makes the weights the mechanism. The Parzen window has a comparable sum and a different shape, and it sits at 0.871: its weights stay near one over the first few lags, and the first few lags are where the information is. A weight sum treats every lag as equally informative and no sample does.

What a window leaves free

A Bartlett window's weights sum to exactly half its width, which is a candidate for what the band costs. Varying the weights without varying anything else says the weights are the mechanism; varying the shape at the same weight says they are not the arithmetic.

dimension · Criterion What a search costs is not a property of that search. The likelihood ratio a searched break in the regression reports, two ways, on every law. On its own — the whole rule being a split of the sample, at no whitening — it averages 34.70 under AR(1) at 0.8, against the 11.07 a chi-square on the five coefficients a split adds would use as a threshold. Inside a rule that also chooses a window from a list of eight, the same search adds only 15.40 — less than half. Most of what a break search finds under correlated errors is the correlation, and a whitening chosen from the same sample has taken it already. A charge measured for one search, carried into a rule that makes two, is not conservative in some harmless direction: it is measuring a different quantity.

A charge that depends on the rule

The break search's charge is 34.7 on its own and 15.4 once a window has been chosen from the same sample. Most of what a break search finds under correlated errors is the correlation, and a whitening has taken it already.

twice · Break point What a longer list actually changes. How often the fifteen candidates choose different tuning parameters, at a true null where every one of them contains the truth, so a disagreement is manufactured rather than discovered. The order's list is an interval of integers, and thinning it moves the rate smoothly from 0% at two values to 39% at thirteen. The window's is not an interval — it runs 0, 1, 2, 4, 8, 12, 20, 30 — so a thinned window list jumps depending on whether it happens to keep the width the criterion wants, between 0% and 42% with no order to it. So "the same length" was never quite the same thing for the two rules, and it is a smaller effect than the field it was invoked to explain.

A list is not a rule

How often fifteen candidates disagree about a tuning parameter runs from nothing at two values on the list to two draws in five at thirteen. What the disagreement costs does not move at all.

lists · Order selection Cut the charge and the width follows it. The band width each charge picks, averaged over 400 draws of 120 rows under AR(1) at 0.8, with the standard deviation across draws beside it. Schwarz's charge — half a log n a lag, which is 2.39 here — picks 3.67. Akaike's picks 6.02. Charging the numbers the window actually leaves free, which is half the width, picks 10.12; charging what the optimism measures, 0.767 of that, picks 14.15. A charge and the width it buys are very nearly reciprocal, which is what a likelihood rising at a fixed rate a lag implies and is why the four answers span a factor of 3.86. The width that was actually best on the draw averages 13.90 and moves by 10.30 from draw to draw — three times as much as any rule's answer does.

A width that moves and an error that does not

Four charges give four widths a factor of four apart and four errors half a per cent apart. The derived charge wins, significantly, by a quarter of what was on offer — and none of the four is an estimate of anything.

dimension · Criterion A wider band is always a better fit. The likelihood maximised over the band, at five widths, averaged over 30 samples. A band at L lags is a band at L + 1 with the last entry held at zero, so the families are nested and the maximised likelihood cannot fall — it does not, on any draw. What it does is rise at 0.984 of log-likelihood a lag. A parameter that is doing nothing buys half a unit in expectation and Akaike's criterion charges one, so this is a criterion very nearly indifferent between every width on offer. The dashed line is what a charge of one unit a lag would exactly cancel. Nothing in the fit chooses a width, and what does choose one is a charge somebody has to pick.

Nothing in the fit picks the width

A wider band is always a better fit, and it is better by about one unit of log-likelihood a lag — which is the order of what a criterion charges for a parameter. Three defensible rules choose widths a factor of three apart.

family · Order selection Every pair of sieve orders. Each cell is a tuning list of exactly two sieve orders, the row's and the column's, and its shade is how often fifteen candidate regressions choose different values from it, at a true null over 240 draws. Every pair holding order zero disagrees on no draw at all, because the criterion never picks zero under this law. Every adjacent pair of positive orders disagrees on between 17% and 25% of draws — 22.1% for one and two, 25.0% for eleven and twelve — and the rate falls as the two orders move apart, to 5.0% for one and twelve. The largest cell is 25.0%.

Two values the criterion cannot tell apart

Fifteen candidates disagree about a whitening's order on two draws in five when the list holds all thirteen orders — and on more than one in five when it holds only one and two. Every adjacent pair of positive orders disagrees about as often as that pair does, and every pair holding an order the criterion never picks disagrees on none. A list's disagreement is set by the pairs of values on it that the criterion cannot tell apart, not by how many values it has, and one candidate's own doubt about its pick predicts all fifteen candidates' disagreement with a rank correlation of 0.96.

lists · Order selection How often two units contain a break that is there. 300 draws at each of 5 break sizes, a shift from row 60 of 120 on first-order autoregressive errors. The share of draws on which the profile's two-unit set contains the true row: at the fixed, unwhitened window 12.0% at 0.3, 19.0% at 0.6, 31.7% at 1, 51.7% at 1.5, 69.3% at 2; with the window searched 22.7% at 0.3, 30.0% at 0.6, 44.3% at 1, 63.3% at 1.5, 75.3% at 2. A 95% interval would sit at the top of the frame.

A break that is there

Put a real break in the regression at row 60 and the profile's two-unit set still misses it on most draws — it contains the true row on 12% of draws at a small break and 69% at a large one, without whitening; with the window searched as well, 23% to 75%. A set that would contain it on 95% of draws needs a threshold of about 19 units without whitening and about 8.5 with it, and either honest set holds most of the sample: fifty-seven of the seventy-three candidate rows at a break of one standard deviation. Searching the window does not move where the break is found.

twice · Break point Which factor the sample settles. The field's decomposition at the full tuning list, by sample size, over 1,200 draws at each: how often fifteen candidates choose different values, what share of those disagreements change which candidate the table selects, and the product — how often the list changes the winner. The window: 120 rows 43.3%, 28.5%, 12.33%; 240 rows 17.5%, 14.8%, 2.58%; 480 rows 38.3%, 2.6%, 1.00%; 960 rows 11.6%, 2.2%, 0.25%. The order: 120 rows 48.9%, 21.6%, 10.58%; 240 rows 47.5%, 13.2%, 6.25%; 480 rows 46.1%, 5.1%, 2.33%.

What a longer sample settles

At a hundred and twenty rows, letting each candidate choose its own whitening changes which of fifteen candidates wins on about an eighth of draws, whatever the list, the table or the law. At four hundred and eighty rows it changes the winner on 1.0% of draws for a window and 2.3% for an order, and at nine hundred and sixty on a quarter of one per cent. The sample settles it through one factor only: how often the candidates disagree about the tuning value hardly moves, and the share of their disagreements that change the winner collapses.

apiece · Order selection What a lag of band costs, as the sample grows. The optimism one more lag of a Bartlett band adds, measured by fitting on one sample and scoring on an independent one, 2,400 pairs at each size, with two standard errors: 0.3745 ± 0.0176 at 120 rows, 0.3842 ± 0.0266 at 240 rows, 0.4523 ± 0.0392 at 480 rows, 0.4022 ± 0.0563 at 960 rows. The window's summed weights give one half a lag and the convention charges one.

The charge at four times the sample

A band of lags is charged a full unit apiece and is worth about three eighths of one at a hundred and twenty rows. The field's last essay said the error in that convention was harmless only because the error surface is flat, and that a steeper surface would give it a cost. A longer sample was the obvious place for one. It is not there: from 120 rows to 480 the error's spread across band widths falls by three quarters, and the derived charge's advantage over the convention falls from 5.5 standard errors to 1.7. What the sample does move is the charge itself, from 0.37 of a unit a lag towards the half a unit the window's weights predict.

dimension · Criterion Past the cap, long memory's surface is flat. The prediction error at every band width from four lags to eighty, above the lowest on the extended list, under long memory at d = 4/9 at three sample sizes, 600 draws each. At 120 rows the floor is at 12 lags and eighty is 0.0039 above it. At 480 the floor is at 30 lags and every width from twenty-four to eighty is within 0.0003 of it. At 960 it is at 48 and thirty to eighty lags are within 0.0000. The dashed line is the list's cap: the surface does not rise behind it, it levels off.

A cap that decides nothing

Every band-width list this field priced stopped at thirty lags, and long memory at a large sample is the one dependence whose best band wants to be wider. At 960 rows it does: the draw's own best width lies past the cap on 48.7% of draws. Lift the cap to eighty and the measured charge follows, from 28 lags to 46 — and the error moves by two ten-thousandths, in the wrong direction and inside its standard error, because the surface behind the cap is flat. The field's last conditional fails where it had its best chance.

dimension · Criterion When a longer sample settles the tuning list. How often the tuning list changes which candidate the table selects, against the sample size, for the whitening window (solid) and the sieve order (dashed), on three tables of candidates, twelve hundred draws at each size. Every subset of four predictors: window 12.33% at 120, 2.58% at 240, 1.00% at 480, 0.25% at 960; order 10.58% at 120, 6.25% at 240, 2.33% at 480. The nested sequence of four: window 5.83% at 120, 1.25% at 240, 0.67% at 480, 0.25% at 960; order 5.83% at 120, 4.00% at 240, 2.25% at 480. Six subsets of two predictors each: window 11.75% at 120, 3.75% at 240, 5.75% at 480, 2.33% at 960; order 9.42% at 120, 6.67% at 240, 4.42% at 480. The nested table starts at half the full table's rate and reaches a small one a sample size sooner; the table whose candidates all have the same dimension is still changing its winner on one draw in forty-three at 960 rows.

The table decides when the sample settles it

Letting each candidate regression choose its own whitening changes which candidate wins on about an eighth of draws at 120 rows, and a longer sample settles it — for fifteen nested-and-crossed subsets, at 0.25% by 960 rows. A nested sequence of four candidates gets there a sample size sooner, from half the rate. Six candidates that all hold two predictors never get there: at 960 rows their tuning list still changes the winner on 2.3% of draws, nine times the full table's, because a disagreement among candidates of one size goes on deciding the winner as often at 960 rows as at 240.

apiece · Order selection What a quarrel between two orders costs. Every two-value list of autoregressive orders from one to twelve, read on the same 2400 draws of a first-order autoregression at a hundred and twenty rows: each cell is the average regret of choosing the order per candidate rather than once for the table, on the draws where the fifteen candidates disagree about it. The cost grows away from the diagonal and stays level along it. Next to the diagonal, a quarrel between neighbouring orders costs 0.00126 for one and two and 0.00088 for eleven and twelve; four orders apart it costs 0.00803 and 0.00802; eleven apart, 0.02436. Every list holding order zero is left off because the candidates never disagree about one.

A quarrel's cost is its width

When fifteen candidate regressions disagree about a whitening's autoregressive order, the disagreement costs in proportion to how far apart the two orders are — 0.0027 of regret for every order of distance, a line that explains 96% of the cost across sixty-six pairs — and not at all by which orders they are: a quarrel between eleven and twelve costs what one between one and two costs, though one of those holds the truth. Fitted on pairs, the line predicts every longer list's cost. The window's quarrels follow no such line.

lists · Order selection The floor a repair has to choose. Regret of the truncated covariance repaired by flooring its eigenvalues, at a persistence of 0.85 on 120 rows, 120 draws, by floor, on a log scale. Window 4: 1.5649 at 0.001, 0.0945 at 0.01, 0.0385 at 0.05, 0.0285 at 0.1, 0.0226 at 0.2, 0.0235 at 0.4, 0.0292 at 0.7; the taper at the same window 0.0275. Window 8: 1.1919 at 0.001, 0.0980 at 0.01, 0.0334 at 0.05, 0.0258 at 0.1, 0.0197 at 0.2, 0.0209 at 0.4, 0.0249 at 0.7; the taper at the same window 0.0178. Window 12: 0.8692 at 0.001, 0.0621 at 0.01, 0.0239 at 0.05, 0.0209 at 0.1, 0.0212 at 0.2, 0.0239 at 0.4, 0.0282 at 0.7; the taper at the same window 0.0150. Window 20: 1.7110 at 0.001, 0.2495 at 0.01, 0.0399 at 0.05, 0.0238 at 0.1, 0.0202 at 0.2, 0.0219 at 0.4, 0.0259 at 0.7; the taper at the same window 0.0135. Least squares gives up 0.0890 and the whitening told the truth 0.0075.

A covariance repaired

A truncated autocovariance sequence fails to be a covariance matrix on most draws of a persistent series, and this line of essays has always scored it on the draws where it survives. The textbook repair is to floor its eigenvalues. At a floor of 0.001 that moves the matrix by four per cent and gives up 1.56 of regret, seventeen times what least squares gives up, because a whitening reads an inverse and an inverse reads the smallest eigenvalues. At a floor of 0.2 it beats the taper at the shortest windows. So the repair is not a repair: it is a second dial.

banded · Dependence

Named alongside it

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

Information criterionModel selectionMonte CarloWhiteningSelection effectRegretAutoregressionDependenceTaperingCovariance matrixDegrees of freedomNuisance parameter

All concepts