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 9 essays across 4 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 five 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 five 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

Named alongside it

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

Information criterionDegrees of freedomModel selectionCovariance matrixDependenceMonte CarloNuisance parameterSelection effectTaperingWhiteningRegretClosed form

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