Criterion — the series · Normal Distribution

A design is a number

A standard design is taken from a catalogue and then measured. Turn the arithmetic round and a design becomes the answer to an optimisation — and over 121 candidate settings the search keeps nine of them, which are exactly the nine a catalogue would have offered, at weights nine equal runs cannot express.

part 1 · optimality

Four criteria on 5 designs, on a square region. Each design scored as an efficiency — its value over the best attainable — so four criteria in as many different units sit on one scale where 1 is the optimum. Rows are ordered by D. D picks 13-run exchange; A picks face-centred composite; G picks 13-run exchange; I picks face-centred composite. Every design has been scaled to just fit the region first, because a design run at settings the region does not contain is not a competitor on it. The disagreement is the point: the letter is a choice, and it is almost never reported as one.

Four letters and two camps

D, A, G and I are four ways of turning one matrix into one number, and they do not agree. The design that wins on D is the worst thing here on I. And the same two designs swap places entirely when the region changes from a square to a disc — on all four criteria at once.

part 2 · optimality

Adding a run can make the design worse. The D-efficiency of the best N-run design at each size, against the optimal measure. It is not a rising curve. 13 runs reaches 99.77% and 14 falls to 99.44%, because the optimal weights are real numbers and N runs is an integer approximation to them, so how good a design can be depends on how well N divides. A Wald interval behaves the same way: a larger sample sometimes makes its coverage worse, for exactly this reason.

The design that has to be integers

The optimal design is a set of real weights and an experiment is a set of runs, so the theory's answer is never available. Thirteen runs reach 99.77% of it and fourteen reach 99.44% — adding a run makes the design worse per run, and the search that finds it does not always find the same one.

part 3 · optimality

The end of the family is the member the algorithm cannot reach. Every member of the Φₚ family optimised on the same 11² candidates, and two eigenvalues of each answer. The upper curve is the smallest eigenvalue of the information matrix — the quantity E-optimality maximises — which rises from 0.0993 at the D end to 0.1994 at p = 64. The lower curve is the gap between that eigenvalue and the next one up, which falls from 0.0611 to 0.0011. A smallest eigenvalue is not differentiable where it is repeated, and the family is driving the gap to zero: the one criterion here whose meaning fits in a sentence is the one whose optimum sits on a corner of its own surface. The candidate grid is on a slider and it answers a narrower question than it looks. On this square region, refining an odd grid from seven to eleven moves nothing at all — the optimum's support is the corners, the edge midpoints and the centre, and every odd grid from five up contains all of them. An even grid has no centre point and cannot reach the answer at any member of the family. On a disc, where the boundary passes through no grid point, refinement does move it.

The family behind the letters

A, D and E are not three ideas. They are three points of one family with a single dial, and running the dial from one end to the other doubles the smallest eigenvalue of the information matrix while closing the gap above it fifty-three-fold — which is the family driving its own last member to the place where it stops being differentiable.

part 4 · criteria

The two terms anybody wanted

D-optimality estimates all six parameters of a quadratic as precisely as possible. Nobody wants that. An experimenter looking for a maximum wants the two curvature terms, and the design that gives them is not the D-optimal one — it is a quarter of the runs at the centre, exactly, and the D-optimal design is 75.3% efficient for the question that was actually asked.

part 5 · criteria

Two designs for one model, and the weights are not equal. Where the runs go, for the same two-parameter model at K = 1 and a ceiling of T = 10. The D-optimal design for both parameters is the familiar one: half the runs at 0.8333 and half at the ceiling. The design for the half-saturation constant alone moves the lower setting down to 0.6040 — where the response curve is still bending, which is where K is visible — and, unlike every design in the two fields before this one, it does not split the runs evenly: the weights are exactly 1/√2 and 1 − 1/√2, 0.7071 and 0.2929, at every K, V and T. The equal weights of D-optimality were a consequence of asking about both parameters at once, and nobody had to notice while that was the only question being asked.

An efficiency that is a ratio

A design chosen for a model is not a design chosen for the parameter somebody wanted. Asking for one of two parameters moves the runs, unbalances the weights, and costs the other question exactly 15.07% — at every setting, because it is algebra.

part 6 · guarantee

What the main effects cost the rest of the model. The compound optimum of α·log Ds(main effects) + (1 − α)·log D on an 11-by-11 grid of the square, at 14 weights from α = 0 to 0.999, placed by its D-efficiency against the D-optimal design and its main-effect efficiency against the factorial. The D-optimal design (α = 0) gives the main effects 74.3% of what the factorial gives. At α = 0.5 the design keeps 94.6% of D and gives 85.6%; at 0.9, 64.9% and 96.8%; at 0.99, 31.3% and 99.7%. The factorial itself (open, at the left edge) gives the main effects everything and the model nothing: its D-efficiency is zero. The design optimal for the two quadratic terms (red) gives the main effects 50.0% at a D-efficiency of 83.6%.

A subset the model cannot use

Nominate the two main effects of a quadratic response surface as the parameters of interest and the optimal design is the 2² factorial — four corners, a quarter each — which leaves the full model's information matrix at rank four of six and the curvature with nothing at all. That is why the field's algorithm would not converge on it. Approached through a compound criterion instead, the trade is explicit: 96.8% of the main effects' best information costs the model 35% of its D-efficiency, and the last three points cost nine tenths of what is left about the curvature.

part 6 · criteria

Four designs, and what each of them guarantees. The worst Ds-efficiency each design achieves anywhere in a rectangle of parameter values 4 times wide in each coordinate. The design built at the guess guarantees 7.3%: it is perfect where it was built and nearly useless at one corner. The D-optimal design at the same guess guarantees 25.6% — it answers the wrong question everywhere and is therefore not concentrated on being right anywhere. The third is the one this field exists to measure: maximin over the first parameter's whole range, with the second held at its guess. It guarantees 7.0%, which is no better than the design that protects nothing. Protecting both is worth 45.9%, and it needs 3 settings to do it.

The worst case in two directions

A design that protects a range of one parameter is robust. Protect the range of one parameter while holding the other at a guess and the design is still robust, still has a guarantee, and guarantees no more than a design that protects nothing at all.

part 7 · blind

What a stated power leaves of the main effects. The main-effect efficiency of the best design reaching a power of 0.8 at level 0.05, as N|δ|² rises through 50, 60, 70, 80, 100, 130, 170, 220, 300, 400, 600, 1000. a saddle, two-degree test, the compound path: out of reach, out of reach, 0.806, 0.841, 0.881, 0.914, 0.937, 0.952, 0.966, 0.975, 0.984, 0.990. a bowl, two-degree test, the compound path: 0.782, 0.823, 0.851, 0.871, 0.898, 0.923, 0.941, 0.955, 0.967, 0.976, 0.984, 0.990. a bowl, its own one-degree contrast, the compound path: 0.827, 0.859, 0.880, 0.896, 0.918, 0.937, 0.952, 0.964, 0.973, 0.980, 0.987, 0.992. a bowl, a factorial with centre points: 0.914, 0.930, 0.940, 0.948, 0.959, 0.969, 0.976, 0.982, 0.987, 0.990, 0.993, 0.996.

A design chosen for a test

A screening experiment that wants its main effects and a check on curvature has been told to trade them with a weight nobody can state. It can state a power. Read against a power of 80%, the frontier collapses to one curve in runs times the squared size of the curvature, and a large enough trial pays almost nothing. But the curve belongs to a test. A factorial with a few centre points beats the entire frontier on a bowl, spending 4% of its main-effect information where the frontier spends 10%, and against a saddle of the same size it has exactly its 5% level.

part 7 · criteria

What a rule reads, against what the outcome uses. The variance of the treatment estimate relative to a coin's, for four things a rule might balance against three shapes the outcome might have, over 350 trials of 200 units. The diagonal is the easy part — a rule that reads the function the outcome uses removes about half the variance. What the table is for is the off-diagonal: reading the covariate alone is worth nothing against a quadratic (0.755), and reading all three is worth nearly as much against every shape as the matching rule is against its own (0.532, 0.493, 0.514).

Three functions of one number

A rule that balances the covariate is exposed to every shape the outcome might have. A rule that balances three functions of it costs two points of variance against the shape the first was built for and takes the worst case from a coin's to about half of it.

part 8 · shape

A basis is a subspace

A balancing rule cannot tell one basis from another with the same span, so choosing what to hand it is choosing a subspace — and then what it removes of any outcome shape is a projection, computable exactly, with no trial anywhere in it.

part 9 · basis

The zero was a fact about independence. What a balancing rule handed every main effect of both covariates removes of a pure interaction, as the covariates are allowed to move together. At ρ = 0 it is exactly nothing — at machine precision, at any number of main effects — which is the independent-covariate result and is correct. It is not small anywhere else: the product of the two covariates loses 64.0% of itself by ρ = 0.5, because h₁h₁ = h₀ + √2·h₂ and Mehler pairs h₂ with h₂ at ρ². Four interactions are drawn and none of them keeps the zero.

A zero that was an assumption

A rule handed every main effect of both covariates removes exactly none of a pure interaction. That is true at machine precision, it is a fact about independence, and it dies as the square of the correlation.

part 9 · joint

How many functions a trial can hold

A rule that reads more functions of one covariate protects more shapes, and every function it reads costs the shapes it already held: the imbalance it leaves in a linear outcome falls like one over the trial size, with a constant that runs from about 3 for one function to about 16 for six. At fifty units that cost eats the protection, and the worst case over four shapes sits between 37% and 42% of a coin's from two functions to six. At four hundred it falls from 42% to 6%. By six functions the worst shape, at every size, is one the rule is reading.

part 9 · shape

The guarantee that survives a correlation, and the one that does not. What a balancing rule handed both main effects removes of the pure interaction between them, as the covariates become dependent. For median splits it is exactly zero at every correlation, because sign(x)² = 1: the interaction sign(X)sign(Y) is orthogonal to sign(X) and to sign(Y) whatever ρ is. For the product of the raw covariates it is 4ρ²/(1+ρ²)² — 64.00% by ρ = 0.5, rising to all of it at perfect correlation. A cut away from the median sits between them and is not small: 23.01% at a cut of one. The zero is not a fact about interactions. It is a fact about a dictionary whose functions square to a constant, which a polynomial one does not.

The zero that survives a cut

A rule holding both main effects removes half of a pure interaction between correlated powers and exactly none between correlated median splits. The guarantee that a correlation destroyed was never about interactions.

part 10 · splits

The share a sample removes

A rule balancing eight main effects of two independent covariates removes exactly none of their interaction — in the population. A trial is a sample, and in a sample of a hundred units exact balance on the same eight functions removes 24% of the interaction's variance on average, between 10% and 42% from one trial to the next. At twenty-five units it removes 62%. The zero is the centre of nothing: it is the limit the in-sample share falls towards, by about two fifths with every doubling of the trial, carried there by an interaction whose spread grows with the very functions it is regressed on. At a correlation of a half the closed form says 64%; a trial of a hundred gets 69% on average and anything from 53% to 83%.

part 10 · joint

The list it was chosen on

Six functions of one covariate looked like 720 designs, one per order of addition. The rule reads a span and not a sequence, so they are 63, and at a hundred units the best four leave 12.1% of a coin's imbalance against the designer's first four at 31.5%. Those best four contain no cube and are, function for function, the four shapes the worst case was scored on. Read on six shapes it was not told about, the same design leaves the cube 77.0% of a coin's imbalance, and the designer's four, which lost by a factor of two and a half, leave it 28.1%.

part 10 · shape

The rule is parity, and it runs both ways. At a correlation of 0.5, four combinations of a dictionary and an outcome shape. The joint sign flip (X, Y) → (−X, −Y) leaves the bivariate normal alone at every correlation, so a function that changes sign under it is orthogonal to one that does not. A product of two odd functions is even; a product of an odd and an even one is odd. So an odd dictionary removes exactly none of the first and something of the second, and an even dictionary does the reverse — which it does, to machine precision, in both of the two rows that should be zero. This is one rule where there had been two: that a median split's square is constant, and that a polynomial dictionary contains the products a correlation generates.

A dictionary that is neither

A rule handed two median splits removes none of their interaction; a rule handed two covariates removes none of their product. Those were two results with two explanations, and they are one result with one — and finding it corrected the number underneath both.

part 11 · dict

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.

part 12 · dimension

Two searches that share nothing

Two searches over independent columns remove shares of the residual sum that add exactly. On the scale a chi-square point is quoted on they look super-additive by a fifth of a unit, and none of it is overlap.

part 12 · apart

What the extra function buys

A rule balancing the mean of each covariate has a worst case of exactly zero. Adding the median split — the other thing every trial balances — leaves it at exactly zero, and one square moves it.

part 12 · dict

One zero holds and one does not. Three rules, at a correlation of 0.5, against the skewness of the covariate. A rule balancing the mean of each covariate removes exactly nothing of their product when the marginal is symmetric — including the heavy-tailed symmetric one at skewness zero, which is what says the guarantee needs symmetry rather than normality — and removes up to 29.7% when it is not. A rule balancing a median split of each removes exactly nothing of the product of the splits under every marginal here, to 1e-30: both sides are functions of the sign of the latent normal, and a monotone transformation moves neither. A rule balancing a threshold at a value on the covariate's own scale removes between 4.9% and 22.5% — it never had a zero to lose, under any marginal at all.

A zero that rests on a symmetry

A balancing rule removes exactly none of an interaction between two odd functions, at every correlation. The argument needs the joint sign flip to preserve the law, and no real covariate is symmetric about anything.

part 13 · skew

A search that is already the other

A break search shifts every coefficient after a row, so a step column is one of the directions it can move in. Paired with a dictionary of them it reads exactly one, on every draw, and that fixes the top of the scale.

part 13 · apart

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.

part 13 · dimension

Two functions a covariate

Every constraint costs admissible assignments at the same rate, so the decision a trial faces is a budget: which two functions of each covariate. Over twenty-eight pairs the worst case never picks the mean. At a correlation of a half it picks the square with a cut at one, beating the mean and the square by sixteen hundredths of a point on a floor held by a shape no pair removes a tenth of — and giving up 41% of a linear outcome to do it. Over the 127 lists of shapes a designer could write, twenty-three different pairs are best somewhere.

part 13 · dict

One zero is arithmetic and one is a symmetry. What two balancing rules remove of the interaction they are aimed at, on five joint laws of the ranks matched at a Spearman correlation of 0.4, with a normal covariate throughout. A rule holding a median split of each covariate removes exactly nothing of the product of the splits under every one of them, including the two that are not symmetric under reflection — and the reason is not a symmetry at all: a centred median split takes the values ±½, so its square is a quarter identically, and the interaction is orthogonal to both main effects whatever the joint law is. A rule holding the mean of each removes exactly nothing under the three radially symmetric copulas and 7.71% under the two that are not. Bars at the floor are exact zeros; the axis cannot draw 9e-32.

A zero that is arithmetic

A median split's exact zero was explained by a symmetry of the latent normal. It holds under a Clayton copula, which has no such symmetry, because a centred median split squares to a quarter identically.

part 14 · copula

Where the two kinds of cut sit. Six covariates, each a monotone transformation of the same latent normal. The vertical line at zero is where every median split sits, on every one of them, because a monotone map preserves order: the median of the covariate is the image of the median of the latent normal. The marks on the curves are where a threshold at 1 on the covariate's scale falls — 1.000, 0.881, 0.875, 0.783, 0.713, 0.337 — and none of them is at zero. That is the whole of the difference. A function of the sign of the latent normal is odd, and a rule made of odd functions removes exactly nothing of an interaction between two of them; a threshold anywhere else is neither odd nor even and removes something.

A split survives what a mean does not

The two things every trial balances come apart on a skewed covariate. A median split is a function of the sign of the latent normal whatever the marginal is; a mean is not, and its exact zero is gone at a skewness of one.

part 14 · skew

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.

part 14 · dimension

Three quarters of the way to one search

The pair that started this reads 0.762 on a scale whose one is containment. And the pair that shares nothing but its response reads −0.306, so the sign the earlier field found does not transport at all.

part 14 · apart

The mean's zero is the copula's symmetry. Five copulas, each at a Spearman rank correlation of 0.4, with a normal covariate throughout — so nothing here is about the marginal, which is the whole of the earlier field. Horizontally: how far the copula's density is from its own reflection through the centre of the unit square, measured rather than read off the family's name. Vertically: what a rule balancing the mean of each covariate removes of their product. The three copulas at zero on the horizontal axis remove exactly nothing, to thirty decimal places. The two that are not symmetric remove 7.71%. A guarantee that held for six marginals turns out to have needed something the marginals could not have told anybody about.

The symmetry the marginals could not show

A mean's interaction zero needs the covariate to be symmetric and the copula to be symmetric under reflection. Six marginals could only ever test one of those, and the other is broken by the commonest kind of dependence there is.

part 15 · copula

A cut at a quantile, and a cut at a value. Two rules that read identically in a protocol. One splits each covariate at its median; the other splits it at 1 on the covariate's own scale — a dose, a temperature, a clinical threshold. At a correlation of 0.5 the first removes exactly nothing of the interaction between its own two splits, under every marginal here, because a median split is a function of the sign of the latent normal whatever the marginal is. The second removes what the bars show, and it does so on a normal covariate too: the threshold sits at 1.000 on the latent scale rather than at zero, so it is 59.4% odd and 40.6% even. The exact zero was never about the cut; it was about the cut being at the median.

The cut that is not a quantile

A protocol that says split the covariate at a threshold and one that says split it at the median read the same and are different rules. One has an exact guarantee under every marginal and the other has none under any.

part 15 · skew

The overlap at the critical value

Two searches over independent noise columns are additive on average: the joint search finds what the two find separately, to within a fortieth of the smaller. At the point a test actually uses — the joint statistic's 95% quantile — the same pair overlaps by 0.320, and a rule that adds the two searches' own critical values rejects 1.4% of true nulls instead of 5%. Nothing about the searches causes it. Quantiles of a sum do not add, and on every pair in the field the overlap at the critical value is larger than the overlap the field measured.

part 15 · apart

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.

part 15 · dimension

Which tail the threshold is in. What a rule balancing a threshold at 1 on each covariate's own scale removes of the interaction between the two thresholds, on five copulas matched at a Spearman rank correlation of 0.4 with a normal covariate throughout. This rule never had a zero to lose — the earlier field establishes that under every marginal — so what is left is a size, and the size depends on where the dependence lives. A Clayton copula, whose density piles up in the lower tail, leaves 5.33%; the same copula turned over, so that it piles up in the upper tail where the threshold is, leaves 33.36%. Same rank correlation, same Kendall tau, same marginal, same threshold: 6.26 times the leak, decided by which end of the distribution the dependence and the cut are both in.

Which tail the cut sits in

The same copula and its reflection have the same rank correlation, the same Kendall tau and the same marginals. A balancing rule holding a threshold at a dose leaves 5.33% under one and 33.36% under the other.

part 16 · copula

A guarantee that stops being a number. The worst case of each dictionary over six outcome shapes, at a correlation of 0.5, against the skewness of the covariate. Under a symmetric marginal every rule made of odd functions has a worst case of exactly zero, and the rule holding a mean and a median split of each covariate — the two things every trial balances — is one of them. Under skew that zero becomes 0.74%, 1.83%, 2.24%, 2.49%: small numbers, each of which depends on a marginal nobody stated. The guarantee has not improved by becoming positive. It has stopped being a guarantee, because it can no longer be written down without the covariate's distribution in it.

Balancing a skewed covariate

The worst case of the rule every trial runs goes from exactly zero to somewhere between a quarter of a per cent and two and a half. Which is small, and is a number that cannot be stated without the covariate's distribution in it.

part 16 · skew

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.

part 16 · dimension

Four rules for a break and a column. How often each of four rules says something is there, for the pair of searches over a break and a column, under four truths, 400 draws each under a first-order autoregression at 0.8 with every threshold simulated on 1200 null draws. nothing there: joint, calibrated 6.3%, summed thresholds 6.3%, Bonferroni 6.3%, either, uncorrected 12.0%. a shift from row 60: joint, calibrated 65.3%, summed thresholds 64.0%, Bonferroni 55.0%, either, uncorrected 69.3%. a column enters: joint, calibrated 11.8%, summed thresholds 10.8%, Bonferroni 73.0%, either, uncorrected 82.5%. both: joint, calibrated 70.5%, summed thresholds 69.0%, Bonferroni 75.8%, either, uncorrected 84.8%. The first block is each rule's size.

The power a calibration buys

Charging two searches separately makes a joint test conservative, and calibrating the joint statistic on its own null repairs that. Against a real break, on the pair of a break search and a window search, the repair is worth 47 points of power — 62.0% against 14.5%. On the pair of a break search and a column search it is worth nothing, and worse: a real column is found by the calibrated joint test on 11.8% of samples and by a plain Bonferroni split on 73.0%, because a joint statistic takes its threshold from the noisier of its two searches.

part 16 · apart

A cut walked from the median into each tail. What a rule balancing a threshold on each of two normal covariates removes of the interaction between the two thresholds, with the cut moved from the 1st percentile to the 99th, on five copulas at a Spearman rank correlation of 0.4. Every copula reads exactly zero at the median. The three radially symmetric copulas are mirror images about it: the Gaussian peaks at 19.49% near the 95th percentile and is down to 13.44% at the 99th; the Frank peaks at 13.22% near the 85th and falls to 2.06%; the t is still near its highest, 37.45%, at the 99th. The Clayton climbs to 56.37% at the 1st percentile, in the tail its dependence is in, and never passes 5.39% in the other; its reflection is the same curve turned round.

A cut walked into the tail

Move a balancing rule's threshold off the median and the interaction it leaks rises from exactly zero — and then either turns back to nothing or climbs to a ceiling, depending on one property of the dependence. With both covariates cut, the whole regression collapses to two probabilities, and far out it settles on 2λ/(1 + λ) of the tail-dependence coefficient: 57.25% under a Clayton, 35.57% under a t, and zero under a Gaussian or a Frank however strongly the covariates are related.

part 17 · copula

A share at a hundred units

On a skewed covariate the closed form credits a rule that balances the means with removing 26% of the covariates' product and 52% of a square, shares that rest on symmetry having failed. A trial does not collect them. At a hundred units the rule removes 12% of the product and 37% of the square on the clearly skewed covariate, and at four hundred units barely more, because the shares live in the long tail and the tail is a handful of units. On the most skewed covariate the rule removes a fifth of the median split at a hundred units and nothing of it at four hundred: balancing the mean of such a covariate is balancing its largest few values.

part 17 · skew

The term that cancels, and the term that does not. The volume each candidate's whitening moves — log|Ω̂| — for a sieve of order 4 on one sample of 120 rows. Estimated once from the fullest candidate and used for the whole table, it is the same number for every candidate, so it drops out of every difference the criterion reads: that is why nothing in this collection has ever needed to carry it. Estimated from each candidate's own residuals it ranges over 23.87, which is more than a parameter is worth, and the criteria being compared are then fits made under different error models with no term saying so. The window's rule has carried this term since the estimated-covariance field and the sieve's never had it.

The volume a whitening moves

A sieve's whitening has a determinant and this collection's criterion for it never carried one. Shared across a table the term cancels exactly, which is why nothing ever noticed; used per candidate it is worth more than a parameter and the whole comparison turns on it.

part 18 · lists

A leak read off a cohort, at two cuts. The threshold rule's leak as a cohort of 1,000 units estimates it from its own two-by-two table: the bar is the middle eighty per cent of 600 cohorts' plug-in estimates, the open circle their median and the stroke the true leak, for five copulas at a Spearman rank correlation of 0.4. At the 90th percentile the estimate is centred on the truth and its middle eighty per cent spans about fourteen points: Gaussian 12.8% to 26.4% around 19.2%, t on four 21.5% to 35.7% around 28.9%, Frank 5.4% to 18.7% around 12.2%, Clayton 0.8% to 9.6% around 4.4%, Clayton turned over 35.9% to 49.5% around 43.1%. At the 99th it is not centred on anything: Gaussian 13.1% to 100.0% around 13.7%, t on four 17.5% to 72.9% around 37.4%, Frank 14.4% to 100.0% around 2.2%, Clayton 16.1% to 100.0% around 0.6%, Clayton turned over 37.0% to 75.7% around 56.3%, because on three of the five copulas between two cohorts in five and four in five have nobody past both cuts.

A leak read off a two-by-two table

The share of an interaction a threshold-balancing rule leaves alone needs one number from the data: how often a unit past the cut on one covariate is past it on the other. A cohort of a thousand estimates it to within a dozen points at the 90th percentile. At the 99th most cohorts have an empty cell in their table, and every one of those reads a leak of exactly one hundred per cent — a reading the bootstrap then confirms with an interval of zero width.

part 18 · copula

The covariate on its ranks

A skewed covariate spoils every parity argument a balancing rule relies on, and a rule that balances ranks instead of values looks like the way out, since a rank has the same uniform law on any marginal. The measurement says the skew really does belong to the scale for one kind of shape: a rank holds a median split at exactly a quarter of a coin's imbalance whatever the covariate. But it pays with a cap on any shape linear in the value, at nearly half a coin's imbalance on the strongest skew. And a rule holding values and median splits together holds both, beating the rank on the shape a rank gives up and matching it on the one it gains.

part 18 · skew

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.

part 19 · curve

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.

part 20 · curve

A line in the right width beats two curves. How far each candidate charge sits from the measured optimism across the plateau, in units of each width's own standard error, over 2000 draws. The straight line through the origin in the band's summed weights — which is what the earlier field levies — misses by 0.2382 per width. The same straight line in the pairs the band actually uses, Σ w(k)(1 − k/n), misses by 0.0095. A fitted power law misses by 0.0293 and a fitted decaying rate by 0.0172, both on one fitted constant more. The deferral this field answers asked for a curve; the answer is a line, in a variable with nothing fitted in it.

A line that beats two curves

A deferral asked for a curve. Fitted against the same measurements, a straight line in a variable nobody had to fit describes the plateau better than either curve does with a constant more — and for three windows out of four it does not.

part 21 · curve

The scale moves the width; the curve does not. The band width each charge picks, averaged over 150 draws of 120 rows. The two conventions — a unit a lag and half a log n a lag — pick 6.08 and 3.65 lags. The four charges derived from the measured optimism pick 15.05, 15.60, 15.47 and 15.44, against a best width on the draw of 14.13. So the scale a charge is levied on moves the width by a factor of 4.27 and the shape of the charge moves it by 3.6%. None of the six is an estimate of the draw's own best width: the correlations are -0.006, -0.003, 0.017, -0.001, 0.016, -0.004.

What a better charge buys

Four charges derived from the same measurements pick band widths within six per cent of each other and deliver errors within two per cent of the gap any of them leaves. The scale a charge is levied on decides the width; the shape of the charge decides nothing.

part 22 · curve

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.

part 23 · curve

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.

part 24 · curve

Four laws, one flat floor. The error a band of each fixed width delivers, averaged over 400 draws of 120 rows under each law, drawn above that law's own best single width. From one lag to the best width the error falls by 0.02763 under AR(1) at 0.8, 0.02517 under a five-period moving average, 0.01122 under long memory at d = 4/9, 0.03507 under a break in the persistence; from six lags to thirty the whole surface spans only 0.00269, 0.00445, 0.00255, 0.00521. The best single width is 12, 24, 24, 16 lags. The flattest surface end to end is the one under long memory at d = 4/9, which is the law a steeper surface was expected from.

The surface under every law

A draw-reading charge for a covariance band failed on one law, and the failure was deferred to a law with a steeper error surface. Run under all four laws in the collection, the surface is flat from six lags to thirty under every one, no draw-reading charge tracks anything, and long memory — the law expected to be steepest — is the flattest end to end. What the law moves is the level of the charge, not whether a draw can be read.

part 25 · curve