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

Sample autocovariance — where it appears

The average product of pairs of observations a fixed number of steps apart, which is what every estimate of a dependence is built from. It is biased downwards at finite length because the sample mean is subtracted from a series that has barely enough independent information to estimate one.

Named by 15 essays across 7 fields — each of them below, with the objects they name alongside it.

Four dependences a single parameter cannot tell apart. Every law here is standardised to a lag-one autocorrelation of 0.8, so a rule told the errors are a first-order autoregression finds the same number in all four and has no way of seeing what separates them. The geometric decay is the world in which estimating a covariance rather than naming it was priced, and found to cost. The five-period moving average has 0.200 at the fourth lag and exactly nothing past it, where the geometric law says 0.328 at the fifth. Long memory at d = 4/9 is still at 0.576 by the twentieth lag, where the geometric law has reached 0.012. The break has no autocorrelation function at all: what is drawn for it is the average over the pairs at each gap, which is what a stationary estimate converges to.

A dependence with a shape

Four ways for errors to repeat, all with the same first lag and nothing else in common. A rule told the errors are a first-order autoregression finds the same number in all four, and is right about one of them.

general · Dependence 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 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 The curvature is in the denominator. The optimism a Bartlett band of each width actually costs, divided by that width, on two ways of measuring the width, over 2000 draws at 120 rows. Measured in the weights the band spends — Σ w(k), which is what the earlier field levies its charges on — the reading falls from 0.9528 at two lags to 0.7486 at thirty, so a charge proportional to the summed weights is too dear at one end and too cheap at the other. Measured in the pairs the band uses — Σ w(k)(1 − k/n), because a lag of k is an average over n − k products — the same readings are flat from 4 lags up, at 0.0084 of χ² per width against 0.2359. The correction has no fitted parameter in it: it is a function of the window, the width and the sample size.

The width a band is measured in

A tapered covariance band spends 84% of its own weights at two lags and 74% at thirty. Every charge in the collection is a straight line through the origin in those weights, so it is too dear at one end and too cheap at the other.

curve · Criterion The curvature is in the denominator. The optimism a Bartlett band of each width actually costs, divided by that width, on two ways of measuring the width, over 2000 draws at 120 rows. Measured in the weights the band spends — Σ w(k), which is what the earlier field levies its charges on — the reading falls from 0.9528 at two lags to 0.7486 at thirty, so a charge proportional to the summed weights is too dear at one end and too cheap at the other. Measured in the pairs the band uses — Σ w(k)(1 − k/n), because a lag of k is an average over n − k products — the same readings are flat from 4 lags up, at 0.0084 of χ² per width against 0.2359. The correction has no fitted parameter in it: it is a function of the window, the width and the sample size.

A lag the sample has less of

A sample autocovariance at lag k is an average over n − k products, not n. Count a band's width in the pairs it actually has and the curvature in its charge goes away, on a correction with nothing fitted in it.

curve · Criterion 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 How much memory a fit takes out, candidate by candidate. Under AR(1) at 0.8, the lag-one autocorrelation a candidate's residuals report, computed exactly for each candidate on 200 draws. The upper line is the law at 0.8000. A candidate that is an intercept alone reports 0.7773 — which is exactly what a sample of 120 errors reports, because an intercept annihilates the sample mean and nothing else, and the two arithmetics agree to the last bit. Every predictor after that takes more out, down to 0.7341 at the fullest candidate. That is the collision this field is about: the rule every whitening here uses estimates its nuisance once, from the fullest candidate, so that the criteria stay comparable — and the fullest candidate is the one whose residuals report the least.

The fit that takes the memory out

A candidate's residuals report less dependence than its errors do, and how much less is arithmetic rather than noise. The rule used for a good reason reads the series that has lost the most.

together · Dependence Most of the rise is the optimiser's, and under one law it is not. The rise in log-likelihood from the tapered plug-in to the maximum over the same eight-lag band, beside what the same optimiser produces on a sample generated from the plug-in's own covariance — where the family is correctly specified by construction and there is nothing to find. Under AR(1) at 0.8 the raw rise is 5.72 and the manufactured baseline is 4.79, leaving 0.93 at 1.8 standard errors; under long memory the excess is 0.14, at 0.2. Under the moving average it is 11.87 at 19.4 standard errors, on every draw. The taper is a shrinkage, and it costs nothing where the sequence decays smoothly and a great deal where it stops dead.

The plug-in and the maximum

A tapered covariance estimate sits five and a half log-likelihood units below the maximum of the likelihood it is substituted into. Four fifths of that is what the optimiser would have found if nothing were missing.

family · Dependence 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 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 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 Four sequences, and the rule only ever sees the last one. Under long memory at d = 4/9, four things that are all called the dependence. The law itself is the top line. What a sample of 120 rows reports on average is the second, computed exactly: subtracting a sample mean takes the first lag from 0.800 to 0.538. What a candidate's residuals report is the third, lower again at 0.472, because a fit removes dependence along with signal. The autoregressions are fitted to that third sequence and reproduce it exactly out to their own order — the Yule–Walker equations are solved to make it so — so everything they say past that is extrapolation. At the twentieth lag the law has 0.576, the residuals report 0.006, and an AR(8) extrapolates 0.028.

The order the tail is drawn at

A fitted autoregression reproduces the sample exactly at the lags it was fitted on, so everything it says past them is extrapolation — and the order is the dial that decides how much of it there is.

general · Order-selection Largest where least is needed. What the pairs correction supplies against what each window's measured profile needs, across this field's plateau, over 2000 draws at 120 rows. Both are stated as the multiplicative rise the charge per unit of width has to take between four lags and thirty. What the correction supplies is arithmetic — (1 − μ(4)/n)/(1 − μ(30)/n), where μ is the mean lag of the weight the band adds — and it runs 1.0795, 1.0580, 1.0456, 1.0539 for the four windows. What the measurement needs runs 1.1076, 1.2928, 1.2296, 1.6550. The two orderings are opposite: the plain Bartlett window has the longest mean lag, so it gets the biggest correction, and the flattest profile, so it needs the smallest. They coincide to 0.9746 of each other, and nowhere else does the correction account for more than 85.0% of the fall.

What the correction assumes

A correction with nothing fitted in it repairs one window of four. The reason is that its size is set by where a window puts its weight and the curvature it must repair is set by something else — and for one window at one sample size the two happen to agree.

curve · Criterion Reading the draw changes what is charged, not what is tracked. The correlation between the band width each rule picks and the best band width on the same draw, over 400 draws. The three fixed charges read -0.069, -0.066, 0.012. The three that read the sample read -0.012, -0.019, -0.041. None of the six is distinguishable from nothing. The statistic the first plug-in reads does vary — the draw's own summed squared autocorrelation runs from 2.06 to 10.19 with a mean of 4.36 — so the failure is not that the charge stopped moving. It is that what it moves with carries no information about which width this draw wanted.

A charge that reads the draw

Three charges built to read the sample track the best band width on their own draw at −0.012, −0.019 and −0.041, deliver more error than the fixed rule they are calibrated to, and pick a width half again as variable. The statistic moves; the answer does not.

curve · Criterion

Named alongside it

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

TaperingModel selectionInformation criterionLong-run varianceClosed formDependenceMonte CarloBandwidth selectionCovariance matrixDegrees of freedomOptimismPlug in estimate

All concepts