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

Copula — where it appears

The joint distribution of two variables' ranks, which is what is left of their dependence once each margin has been relabelled. Every joint law splits into its marginals and its copula, and a monotone transformation of either axis changes the first while leaving the second exactly as it was.

Named by 13 essays across 3 fields — each of them below, with the objects they name alongside it.

One margin rises; the other turns over. The two halves of the table's margin across the sweep: a lower-tail copula's own leak with a symmetric covariate, and a covariate skewed at 0.95 under a Gaussian copula. The marginal's leak rises at every step, from 3.727% to 36.056%. The copula's does not: it rises to 9.064% at a Spearman of 0.6 and falls to 8.219% by 0.7. It has to turn over, because at a rank correlation of one the two variables are a deterministic function of each other and there is no interaction left for a split to leak. So the margin of the table turns over before any cell in it does.

A margin that turns over

A skewed covariate's leak grows without limit as the dependence strengthens. A copula's own leak does not — it peaks at a rank correlation of 0.6 and falls. The margin of the table turns over before any cell in it does.

stronger · Adjustment 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.

copula · Criterion 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.

copula · Criterion A near-perfect cancellation, at one correlation. A covariate skewed at 0.75 under a lower-tail copula, at each of 7 rank correlations. The earlier field measures this cell at a Spearman of 0.4 and reads 0.002% where adding the two halves' own leaks gives 16.579% — a cancellation so near exact that it is that field's headline. Across the sweep the same cell reads 0.0084%, 0.0182%, 0.0097%, 0.0015%, 0.0864%, 0.4693%, 1.5327%. Its smallest value is at 0.4, in the interior, and by 0.7 it is 1007.40 times larger. The near-zero is where two curves cross, and they cross beside the one correlation that was measured.

The zero that was a crossing

A cell that leaks 0.002% where adding its two halves gives 16.6% is a field's headline. On a finer grid it passes through zero at a rank correlation of 0.38 — two hundredths from where it was measured.

stronger · Adjustment Four cells change their answer. The four cells of the twenty whose excess changes sign as the dependence strengthens, over 7 recalibrations. Above the line the two failures compound — the cell leaks more than adding the copula's own leak and the marginal's — and below it they cancel. All four start above and end below, and all four are at the two most skewed covariates: skew 0.90 under heavy-tailed, skew 0.95 under heavy-tailed, skew 0.90 under upper tail, skew 0.95 under upper tail. Whether two failures of a dependence compound or cancel is therefore not a property of the pair. It is a property of the pair at a strength of dependence, and a fifth of the table changes its answer inside the range measured here.

An answer that changes

Eleven of twenty cells cancel and nine compound, at one rank correlation. Sweep the correlation and four of the twenty change sides — all four from compounding to cancelling, all four at the most skewed covariates.

stronger · Adjustment 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 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 table swept along the covariate instead. What a rule holding a mean of each covariate fails to remove of their interaction, at each of 4 copulas, as the covariate is skewed further and the rank correlation is held at 0.4. The sweep runs from a symmetric covariate at g = 0 to a skewness of 11.16, and the three settings the earlier table names — g = 0.3, 0.6 and 0.9 — are on it, where this sweep reproduces that table to the last digit. Every row rises and then falls: the lower-tail copula from 7.707% through 0.0015% and back to 5.587%, the upper-tail one to a maximum of 36.213%. So the quantity a trial is exposed to is not monotone in how skewed its covariate is.

The other dial

The table is swept along the strength of the dependence and never along the shape of the covariate. Swept along the shape at a fixed correlation, the same two copulas cross, the same way — and the near-zero cell turns out to be a minimum in both directions at once.

stronger · Adjustment 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 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 Seven products of three splits. The correlation of every product of three centred median splits with every other, at a Spearman rank correlation of 0.4 between each pair of covariates. Under a Gaussian copula a main effect correlates with the other mains and with the triple (0.273), and with no pair at all, because the third moment is 0.000. Under a Frank copula, whose pairs are radially symmetric, the third moment is 0.078 and every other cell fills in: a main correlates 0.082 with the pair it is not in. In both panels a main is exactly uncorrelated with a pair it belongs to, which is the known zero.

The zero that needed two

A rule balancing two median splits removes nothing of their interaction, on every copula, because a centred split squares to a constant. Add a third covariate and the same arithmetic stops protecting the thing it looked as if it protected. The product of three splits is not orthogonal to the mains: at a rank correlation of 0.4 a rule on the mains removes 14.47% of it under a Gaussian copula. And the symmetry that decides the rest belongs to the triple rather than the pair, so Frank's copula, whose pairs are radially symmetric, leaks where Clayton's, whose pairs are not, barely does.

compound · Adjustment 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

Named alongside it

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

Covariate balanceInteractionMedian splitClosed formQuadratureTail dependenceMarginal distributionRank correlationSymmetryGaussian copulaParityThreshold

All concepts