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

Bias-variance — where it appears

The two ways an estimate can be wrong: systematically, in the same direction every time, and by wandering. Almost every dial in this collection trades one against the other, and the setting that minimises either is rarely the setting that minimises the sum.

Named by 14 essays across 9 fields — each of them below, with the objects they name alongside it.

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.

taper · Bootstrap 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.

crossing · Bootstrap 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.

feasible · Bootstrap 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.

readout · Bootstrap 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.

feasible · Bootstrap 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.

crossing · Bootstrap The crossing is in the dependence, not in the split. Regret of each rule as the design and the errors are made persistent at the same coefficient, scored on fresh rows because the closed form assumes exactly what is being taken away. An optimism theorem counts rows; when the rows repeat each other there are fewer of them than there are rows, the penalty is too small for the fit it is correcting, and the criterion starts buying coefficients it should not — its average winner grows from 3.31 coefficients to 3.90. The hold-out never used the theorem and overtakes at ρ ≈ 0.81. Schwarz's criterion, worst of the three on independent rows, is best on repeating ones — its heavier penalty is right for the wrong reason.

Where the two searches cross

The obvious dial between a criterion and a hold-out is how much of the sample to hold out, and moving it never changes the answer. The dial that does is one nobody chooses — how much each row repeats the one before it — and the two rules change places at about 0.81.

proxy · Forecast 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 The window a whitening wants is not the memory of the errors. Regret under a five-period moving average as the tapered estimate is given more lags, over 120 draws at n = 120. The best window is L = 30; the automatic bandwidth is 4 and the error model's own likelihood chooses 9.7 on average. Both land in the same place and both are short, and the reason is the taper: a Bartlett weight at lag k is 1 − k/(L + 1), so a window of 8 keeps 0.556 of whatever the fourth lag carries and a window of 30 keeps 0.871. A window has to be several times the memory before it stops removing the memory. The dashed line is the rule told the errors are a first-order autoregression, which needs no window at all.

The window a whitening wants

Every law here is best whitened by a window several times longer than its own memory, including the one whose memory ends at the fourth lag. The three ways of choosing it from the sample all land in the same place, and it is the wrong one.

general · Order-selection 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.

feasible · Bootstrap The threshold buys accuracy and spends exceedances. The mean squared error of the estimated shape against the threshold, split into the square of its bias and its spread, over 600 records of 2000 readings from a a normal parent. At the 0.9 quantile 199 exceedances are left, the bias is -0.1708, the spread is 0.0701 and the total error is 0.0341. The bias falls as the threshold rises because the exceedances get closer to being generalised Pareto; the spread rises because there are fewer of them. The sum is smallest at the 0.925 quantile, at 0.0340, of which 80.6% is still bias — so even the best threshold on this grid is one where accuracy, not spread, is the binding constraint.

The threshold is a dial

A peaks-over-threshold analysis has one knob, and raising it buys accuracy with exceedances. For a normal parent the error is smallest at the 0.925 quantile and 80.6% of it is still bias there — and both diagnostics practitioners use to set the knob lose to a fixed 0.90 rule, one by a factor of 1.590 and one by 11.881.

extreme · Extremes A bias against a variance, with the answer in between. How wrong one sample's reference distribution is, split into the two things it is wrong by. Sharing the multiplier over more rows keeps more of the dependence and closes the bias from 1.688 to 0.835; every row it is shared over also removes an independent sign from the 101 the sample started with, and the spread of the resulting quantile rises from 1.307 to 2.172. The distance a practitioner with one sample is actually exposed to is the two together, and it is smallest at ℓ = 5.

How long a block a multiplier shares

Sharing a sign over more rows keeps more of the dependence and leaves fewer independent signs to build a distribution from. The bias falls from 1.6885 to 0.8479 and the spread rises from 1.3073 to 2.1716, and the rejection rate walks straight through its nominal level on the way from 11.3% to 1.3%.

proxy · Reference 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.

crossing · Bootstrap Twenty studies of a thousand readings estimate the regression of the extremes: the t, four degrees parent. Each thin line is one study of 1000 readings from the t, four degrees parent: Tweedie's formula with the log-density's slope estimated by a degree-5 log-spline, drawn up to that study's largest reading. The thick line is the share kept by integration over the true score, and the dashed line the correlation, 0.6. The top ten readings of the median study begin at 2.45. Over 400 studies the corrected share kept by the top one per cent averages 0.8190, with a spread of 0.0806, against 0.7842 by integration; a Gaussian kernel averages 0.7914 with a spread of 0.0853.

The slope of a density nobody can see

Tweedie's formula corrects a reading by the slope of the readings' own log-density, and a study has its readings. Estimated from a thousand of them, the correction for the top one per cent beats the correlation's linear rule on 84.0% to 98.0% of studies from heavy-tailed populations and on 75.0% to 81.5% from a bounded one — and costs an error of 0.09 to 0.12 where the population is normal and the rule was already exact. At 250 readings the log-spline loses to the rule it replaces, and at 16,000 the same log-spline gets worse on a power tail.

paradox · Rtm

Named alongside it

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

Block bootstrapClosed formMonte CarloResamplingTaperingLong-run varianceMean squared errorBlock lengthDependenceAutocorrelationSample autocovarianceTuning parameter

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