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

Threshold — where it appears

A cut in a continuous variable above which something changes, whose correlation with the variable itself falls away as the cut moves into the tail. Balancing the covariate's mean removes exactly 2/π of a median split of it and only a quarter of a cut at one, so the same rule protects two thresholds very differently.

Named by 29 essays across 13 fields — each of them below, with the objects they name alongside it.

The six best bases of 2 functions, and what each protects. Every cell is R²(g | span B) — the share of the imbalance in that shape a rule balancing that basis removes — computed from exact inner products between Hermite functions and indicators, with nothing simulated. The rows are ordered by their worst cell, which is the number an experimenter who does not know the shape is exposed to. The best row here guarantees 26.8% against every shape in the list, and the worst of the six guarantees 15.1%: the difference between them is entirely which subspace was picked, at the same cost per arrival.

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.

basis · Criterion The expansion that never terminates. The Hermite coefficients of a median split, in magnitude, against the reference j to the power −3/4, anchored at the first one. Every even order is exactly zero because sign is an odd function, and every odd order is not, so no truncation is exact — where a polynomial of degree d is exact at any order past d. Summed, the tail past J falls like 1/√J: sixty orders still leave 6.6% of the variance outside. That statement is what made a cut dictionary's geometry unavailable in closed form, and it is a statement about the function against itself. What it is not is the accuracy of an inner product between two correlated variables, where every term past J carries a factor of ρ^m as well.

A cut is not a polynomial, and it does not have to be

A threshold's expansion never terminates, which is why a balancing dictionary's geometry was closed for powers and taken to draws for cut points. Conditioning on the second variable closes it for both.

splits · Blocking Four rules, three shapes, and no ordering that survives. The variance of the unadjusted treatment estimate under each rule, as a fraction of the variance a coin gives, over 450 trials of 200 units each. Against a covariate that enters linearly the rule that reads the number nearly halves it. Against a threshold at 1 it removes about a fifth. Against a quadratic every rule here is at or worse than a coin — they are all optimising a criterion that is one over the variance of an estimate in a model this outcome does not obey, and a constraint that helps nothing still costs something. Nothing in a trial says which column it is in.

Balanced on the wrong function

A rule that reads a covariate's numbers halves the variance of the treatment estimate, if the covariate enters the outcome as a straight line. If it enters as a threshold the rule is worth a fifth of that, and if it enters as a curve every rule here is worse than a coin.

shape · Blocking The fourth-order expectation, by two routes. Every inner product in an eight-term slice of the dictionary at ρ = 0.5, computed from the linearisation and Mehler's formula and counted from two hundred thousand draws of a correlated pair. The entries that matter are the ones off the main effects: ⟨f(X)u(Y), g(X)v(Y)⟩ is a fourth-order expectation, which the independent-covariate field could not write down. The worst departure is 1.99 standard errors over 36 pairs, measured in each pair's own error because the entries differ in size by two orders of magnitude.

The fourth moment that was missing

Mehler's formula makes the main effects exact at any correlation and stops there, because the interactions need an expectation of four Hermite functions rather than two. A linearisation turns the four into two, and the whole geometry becomes closed again.

joint · Blocking 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.

skew · Criterion A threshold in the tail is a threshold nothing balances. The share of a coin's imbalance in an indicator 1{x > c} that survives a rule which balances the covariate itself. The smooth curve is 1 − ρ² with ρ = φ(c)/√(p(1−p)), a closed form with no trial in it; the points are counted over 500 trials of 200 units at each threshold. At the median the two agree that about a third survives — the removed share is exactly 2/π — and by two standard deviations 86.9% survives. The closed form is exact in the limit and optimistic by a few points at this many units, because the rule balances the sample's mean rather than the population's.

A threshold in the tail

How much of a threshold's imbalance a balanced covariate removes is a correlation, and the correlation is a closed form. At the median it is exactly 2/π — the same 2/π a median split throws away — and two standard deviations out it is an eighth.

shape · Blocking Sheppard's arcsine, by two routes. Corr(sign X, sign Y) as the covariates' correlation runs from zero to one, drawn twice. One route is a sixty-four-node quadrature of the orthant probability over the correlation — the general construction, which works at any pair of cut points; the other is (2/π) arcsin ρ, which is elementary and works only at the median. They agree to 3.3e-16 at every one of 81 correlations, which is what licenses the quadrature everywhere else. The curve is above the diagonal at small ρ and below it at large: two signs agree with probability ½ + arcsin(ρ)/π, so a correlation of 0.5 gives exactly ⅓ and a correlation of 0.8 gives 0.5903.

The arcsine that closes it, and the error that was overstated

Two median splits of a correlated pair agree with probability ½ + arcsin(ρ)/π, exactly. And the truncation the field was avoiding falls geometrically in the correlation, not algebraically in the order.

splits · Routes The guarantee, as the basis is allowed more functions. The lower line is the best worst case over the six named shapes for a basis of each size, found by scoring every subset of the dictionary — an exact answer, since the problem is finite. One function guarantees 2.3%, which is nearly nothing; three guarantee 59.0% and the basis that does it is the covariate, its square and its cube, with no indicator in it. The upper line is the same problem with the basis drawn rather than fixed, which is worth 2.09 times as much at two functions and 1.32 at three. The two lines converge because a basis large enough to protect everything has nothing left to randomise over.

Which shapes are worth protecting

Choosing a basis by its worst case is a finite problem with an exact answer. The answer has no tie in it, which a maximin optimum is supposed to have — and the tie comes back, along with twice the guarantee, when the basis is drawn rather than chosen.

basis · Blocking The operating characteristic, and the point that minimises harm at 0.10% prevalence. The published pair — 90% sensitive, 95% specific — is the open mark. With a miss costing 100 times a false alarm and a prevalence of 0.10%, the threshold that minimises expected cost sits at 74.7% sensitivity and 98.81% specificity, with a predictive value of 5.9%.

The test is a point somebody chose

A test reported as 90% sensitive and 95% specific is not two properties of a test. It is one property read at a threshold, and the threshold that minimises harm runs from 3.05 standard deviations of the score at a prevalence of one in ten thousand to −0.12 at one in two — 45% of cases detected at one end and 99.9% at the other.

screening · Baserate 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.

skew · Criterion 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.

splits · Criterion 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.

shape · Criterion A class that is a subspace has no guarantee below its own dimension. Each cell is the worst case over every unit-variance function in a class of dimension m, for a rule reading k functions: the smallest squared principal-angle cosine between the two subspaces. Wherever k is less than m the number is zero to machine precision, and that is not a weak guarantee but the absence of one — some direction of the class is orthogonal to the entire basis, and against an outcome in that direction the rule does exactly what a coin does. An experimenter who declines to name the shapes and asks instead to be protected against everything smooth is asking for the cells above the diagonal.

Where the guarantee is exactly zero

An experimenter who declines to name the shapes, and asks instead to be protected against anything in a class, is asking for a number that is not small but zero. Bounding the class is unavoidable, and the two ways of doing it choose different bases.

basis · Randomisation 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.

copula · Criterion 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 What each analysis does at a true null, by shape. Four analyses of the same trials — 500 of them at each shape, 120 units, assigned by the rule that reads the covariate. Every rejection is false. The unadjusted analysis is the one that moves: 1.60% against a linear outcome, where the design removed a great deal that the standard error still prices, and 5.20% against a quadratic, where it removed nothing and the standard error is right. Adjusting holds the level in all three columns, and so does the design's own reference distribution, which needs to be told the rule and nothing else.

The analysis and the shape

An unadjusted analysis after a rule that read the covariate is too cautious — by a third against a linear outcome, by nothing at all against a quadratic. And an adjustment for the wrong function recovers almost none of the precision the right one would.

shape · Randomisation The one zero neither half of the dependence can touch. A median split's interaction leak at all 30 combinations of copula and marginal, on a log scale. Every one is under 10⁻¹⁶ and the largest is 1.74e-20, which is the quadrature's own noise rather than a leak. The reason is arithmetic and it is short: a centred median split takes the values ±½, so its square is a quarter identically — for every unit, on every draw, whatever the covariate's scale is and whatever joint law the ranks have. The interaction is then orthogonal to both main effects by construction, and there is nothing for either half of the dependence to break. Both of the fields this one joins report this zero holding under their own variation; running both variations at once is what establishes that it is not two coincidences.

The zero that survives both

A median split's interaction leak is under 10⁻¹⁶ at all thirty combinations of copula and marginal. It is the only guarantee in the collection that neither half of the dependence can touch.

compound · Adjustment What a companion with a known expectation can remove from the variance of a simulated 5% rate. For a rate of 0.05, a companion used linearly removes at most 22.4% of the variance however well it correlates. At a correlation of 0.9 it removes 18.1%, the companion's own rate at the same cut 38.2%, and the best function of the companion 48.1%; at 0.99, 21.9%, 77.0% and 82.8%. The companion's rate overtakes the linear companion at a correlation of 0.64.

A rate needs a rate beside it

A simulated rejection rate is an average of indicators, and a companion with an exactly known expectation can remove at most φ(c)²/p(1 − p) of its variance when it is used as a straight line — 22.4% for a 5% rate, 7.2% for a 1% rate — however closely it tracks the statistic. The ceiling is the share of a normal's information a cut keeps. Used as its own rate at the same cut, the same companion removes 38.2% at a correlation of 0.9 and 77.0% at 0.99. On the t test's size for lognormal data it was worth 1.43 times the draws used linearly and 2.32 times used as a rate at a matched level.

splits · Routes 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.

copula · Criterion The line is the sample, and the sample is the finding. 900 draws of two independent standard normal causes, with the 453 of them past a threshold of 0.00 marked and the 447 that fall short left pale. In the population the two are independent by construction. Inside the selected sample the correlation is -0.4669 in closed form and -0.5050 counted on these 453 rows, and the least-squares line through them has a slope of -0.545. The mechanism is visible in the picture rather than argued: the threshold removes one corner of the cloud, and a cloud with a corner missing is a cloud whose two coordinates carry information about each other.

The sample is a condition

Two independent standard normals, selected on their sum exceeding its median, read a correlation of exactly −1/(π − 1) = −0.4669 inside the sample. Nothing is measured badly and nothing is missing — and both halves of that split read it, in the same direction, while the population containing both reads zero.

collider · Conditioning A cohort screened once, the top tenth enrolled, and followed up with nothing given — correlation 0.6. 2000 people read once at screening and once at follow-up, with a test–retest correlation of 0.6 and no treatment. The 215 above a cut at the top ten per cent of one reading (1.282 standard deviations) are enrolled. Their mean screening reading is 1.744 and their mean follow-up reading 0.982, a fall of 0.762 ± 0.055 with nothing done to anyone. The closed form for the fall is (1 − ρ) times the truncated-normal mean, (1 − 0.6) × 1.755 = 0.702.

The measurement that got them enrolled

Enrol the top tenth of one screening reading and give them nothing, and they fall by 0.702 standard deviations at follow-up. Measured from a fresh reading taken after enrolment they fall by nothing. Averaging ten screening readings still leaves 0.101, and it takes twenty-one to get under 0.05.

paradox · Rtm Post-stratifying a simulated 5% rate on a companion correlated 0.9: the share of its variance removed, by the number of strata. The best any function of the companion can do is 48.1%; its own rate at the same cut, 38.2%. Strata spent across the band where the conditional probability changes reach 95% of the best at 8 strata; equal-probability strata need 32. At eight strata the two remove 46.8% and 29.9%.

A companion cut into many

A simulated rate can be estimated within bands of a companion whose distribution is exact, and the bands weighted back by their exact probabilities. One band boundary at the target's threshold is the companion's own rate; infinitely many are the best any function of the companion can do, which for a 5% rate at a correlation of 0.9 removes 48.1% of the variance. Eight bands placed where the target's conditional probability is changing remove 46.8%; eight at equal probabilities remove 29.9%, and need thirty-two to catch up. On the lognormal t test's size, thirty-two bands of the logarithms' t statistic were worth 2.99 times the draws and two hundred and fifty-six, 2.14.

splits · Routes A median split of a measured scale. The share of the interaction a rule balancing a split above the median of each covariate removes, when the covariate is measured on k levels of a symmetric binomial, on five copulas at a Spearman rank correlation of 0.4. With an even number of levels nothing sits on the median and the split is exactly balanced: the rule removes nothing, as for a continuous covariate. With an odd number, 3 levels (50.00% on the median): Gaussian 10.44%, Clayton 4.91%, its reflection 18.44%; 5 levels (37.50% on the median): Gaussian 6.39%, Clayton 3.69%, its reflection 10.06%; 7 levels (31.25% on the median): Gaussian 4.58%, Clayton 2.91%, its reflection 6.77%; 9 levels (27.34% on the median): Gaussian 3.57%, Clayton 2.41%, its reflection 5.05%; 11 levels (24.61% on the median): Gaussian 2.92%, Clayton 2.05%, its reflection 4.01%; 15 levels (20.95% on the median): Gaussian 2.15%, Clayton 1.59%, its reflection 2.82%; 21 levels (17.62% on the median): Gaussian 1.53%, Clayton 1.19%, its reflection 1.93%; 31 levels (14.45% on the median): Gaussian 1.04%, Clayton 0.85%, its reflection 1.26%.

A median with ties

The median split's interaction zero survives every marginal and every copula, because a split that is on for exactly half the units squares to a constant. A covariate measured on a few levels cannot be split exactly in half: on a five-point scale three in eight units sit on the median, and a rule balancing 'above the median' removes 6.39% of the interaction under a Gaussian copula and 10.06% under one of the two Claytons. The size is set by the share of units on the median, which is the formula's own factor. And a fair coin for the tied units restores the zero exactly, on every copula.

compound · Adjustment A threshold read along the skew. What a rule balancing a threshold at the value 1 on each covariate's own scale removes of the interaction between the thresholds, against the covariate's skewness, on five copulas at a Spearman rank correlation of 0.4. Gaussian: 16.41% with no skew, 10.07% at a skewness of 11.16; t on four: 22.18% with no skew, 12.11% at a skewness of 11.16; Frank: 13.19% with no skew, 10.14% at a skewness of 11.16; Clayton: 5.33% with no skew, 4.82% at a skewness of 11.16; Clayton turned over: 33.34% with no skew, 17.61% at a skewness of 11.16. Every leak falls, because the value moves towards the median.

Where the skew puts a threshold

The field read the mean-balancing rule along a sequence of skewness and left the threshold rule — a cut at a fixed value on the covariate's own scale — unread. Read, it turns out to be no new curve. Skewing the scale moves the value 1 from the 84th percentile of the latent covariate to the 74th, and the rule's leak is the copula field's cut sweep read at that quantile: it falls on every copula, from 16.41% to 10.07% under a Gaussian. Whether skew and dependence compound or cancel is then a fact about the slope of each copula's curve, and unlike the mean rule's, no cell changes its answer along the sequence.

stronger · Adjustment The worst case against the number of functions, at four trial sizes. The largest share of a coin's imbalance the rule leaves in any of the four shapes, against how many functions it reads, at 50, 100, 200, 400 units. At 50 units: 103.5%, 38.6%, 37.2%, 37.0%, 39.8%, 41.7%. At 100 units: 105.9%, 37.8%, 38.9%, 30.3%, 27.0%, 24.6%. At 200 units: 95.9%, 44.2%, 42.8%, 32.9%, 28.4%, 11.7%. At 400 units: 108.5%, 41.8%, 41.3%, 32.1%, 26.1%, 5.8%. At fifty units every basis from two functions up leaves the worst shape between 37.0% and 41.7%; at four hundred, six functions leave it 5.8%.

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.

shape · Criterion 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.

copula · Criterion How often a trial of 80 shows a Simpson reversal on its strongest baseline variable, against how many cut points are tried. The treatment helps every patient equally. Cut at its median, the variable's table reverses in 4.4% of trials; trying 2, 3, 5, 9 cuts, in 6.4%, 8.7%, 10.8%, 13.2%. Independent chances would give 33.3% at nine.

Five places to cut one variable

A continuous baseline variable has to be cut before it makes a subgroup table, and the cut is chosen. In a randomised trial of eighty where the treatment helps every patient equally, the strongest baseline variable cut at its median shows a Simpson reversal in 4.4% of trials; tried at five candidate cuts, some cut reverses in 10.8%, and at nine, 13.2%. Nine correlated cuts are worth about five independent chances, not nine. Across twenty continuous variables cut at five points each, some table reverses in 32.2% of trials — twice the rate with each variable cut once, and about thirty-one independent chances where the table holds a hundred.

reversal · Simpson A cut on each covariate, every pair of them. The share of the product of two threshold indicators that a rule balancing the two indicators removes, in per cent, at a Spearman rank correlation of 0.4, for cuts at the fifth to the ninety-fifth percentile on each covariate. On the Gaussian the far corner, a cut at the 5th percentile on one and the 95th on the other, is 54.8% against 19.5% with both at the 95th. On the t copula it is 1.1% against 34.5%. Both surfaces have their only zero at the double median and leave the diagonal lower in places: moving one Gaussian cut from the 80th to the 70th lowers the share from 14.0% to 12.1%.

A cut on each side

Every table of the threshold's leak put one quantile on both covariates, which is the diagonal of a surface nobody had drawn. Off the diagonal the copulas swap places: a cut at the 5th percentile on one covariate and the 95th on the other removes 54.8% of the interaction under a Gaussian copula and 1.1% under a t copula, the reverse of their order at the near corner. Walked further out, the far corner has two opposite limits, and which one a Gaussian reaches is decided by whether its correlation is above or below exactly one third.

stronger · Adjustment A run length of 4 makes every exceedance its own cluster. 200 steps of a max-moving-maximum, X(t) = max(0.4·Z(t), 0.3·Z(t−6), 0.3·Z(t−12)) with unit Fréchet innovations Z, whose extremal index is exactly 0.40: one large innovation can put three readings above a threshold, 6 steps apart. The rule marks the 0.9 quantile and 19 readings clear it; 12 of the gaps between consecutive exceedances are exactly 6 steps. With a run length of 4, so that two exceedances 4 or more steps apart start separate clusters, they form 19 clusters, shaded, the largest holding 1. The runs estimator reads 1.000 against 0.40.

The run length a declustering chooses

The runs estimator of an extremal index carries a constant nobody derives. Where a cluster is a run of neighbouring exceedances the constant barely matters; where a cluster's members fall six steps apart, the estimate is 0.9069 at a run length of six and 0.3649 at seven against an index of 0.40, and a run length of four removes under a tenth of the overstatement declustering exists to remove. A rule that reads the run length off the data has the smallest worst error of the three.

extreme · Extremes

Named alongside it

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

Covariate balanceClosed formInteractionBasis functionsMedian splitOrthogonalityCorrelationMonte CarloAllocation ruleModel misspecificationProjectionRandomisation

All concepts