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

Numerical methods — where it appears

The algorithms a computed answer rests on — decompositions, searches, quadratures — treated here as apparatus rather than as subject. Their failures are quiet: a quantile that returns its own bracket and a series that stops a third of the way both return numbers rather than errors.

Named by 11 essays across 7 fields — each of them below, with the objects they name alongside it.

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 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.

skew · Criterion What a mean split leaves, with both halves varying. The share of a mean split's interaction that survives the rule balancing it, at every copula and every marginal, matched at a Spearman correlation of 0.40. The three radially symmetric copulas leave exactly nothing with a symmetric covariate and rise steeply with the skew. The two asymmetric ones start at 7.707% and go opposite ways: the lower-tail copula falls to 0.002% at a skewness of 0.95 — the two failures cancel almost exactly, and a guarantee both fields report as broken is restored — while the upper-tail one climbs to 40.288%. And the heavy-tailed symmetric covariate, which leaks exactly nothing on its own, doubles what the asymmetric copulas leak: 14.229% against 7.707%.

Two failures that cancel

A mildly skewed covariate under a lower-tail copula leaks 0.002% of an interaction where each failure alone leaks eight and seven per cent. Turn the copula over and the same pair compounds.

compound · Adjustment How far apart the two components are, on each probe. The median separation between the two components of the admissible set — the difference in their mean probe values, over the spread inside a component — over the 100 of 200 designs whose set is enumerated and found split. The separating direction carries 10.565 and needs the enumeration. The fourth power as the earlier fields use it carries 1.543; projected off the span the rule balances, 5.080. The design's own leverage, which uses no dictionary and no outcome, carries 3.836. A random direction in the same subspace carries 0.942, and a direction chosen by looking for concentrated structure carries 0.543 — below random, and the one heuristic here that is worse than not choosing at all.

A probe chosen from the design

The design's own leverage aligns with the separating direction four times better than a random direction in the same subspace. The concentrated direction the argument invites is worse than random.

aimed · Randomisation 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 Coverage of four nominal 95% intervals, n = 30. Computed exactly by summing over all 31 possible counts, not simulated. The Wald interval drops to 26.0% and is jagged everywhere; Clopper–Pearson never falls below 95% and pays for it in width.

Two routes to every number

A site about probability that only simulates has one route to each answer and no way to tell a right one from a plausible one. Every important number here is computed twice, by arithmetic that shares nothing, and the two are required to agree.

method · Routes The optimum is a tie, and the tie is at both ends. The maximin design's efficiency across the range, and underneath it the prior that makes the averaged criterion as bad as possible. The efficiency curve is flat to within 2.1 points, and the minimum 78.74% is attained at K = 0.25 and 0.79 and 0.89 and 1.00 and 1.12 and 4.00 rather than at a single value: if it were attained once, the design could be moved towards that value and the worst case improved, so a tie is what having finished looks like. The bars are the least favourable prior's weights, computed by a completely different route — an averaging problem solved under the weighting that hurts most — and it puts its weight exactly where the ties are, reaching 78.63% against the direct search's 78.74%.

Where the minimum is attained

A design that protects a range is finished when its worst case is a tie. That is a checkable property rather than a description, it is why the search cannot climb a derivative, and it is the same corner the criteria field found at the end of the Φₚ family.

robust · Local design One likelihood, three answers. The concentrated Gaussian log-likelihood of one sample of 120 rows under AR(1) at 0.8, as a function of the correlation the errors are whitened at. Three rules put three different numbers on this curve. The two-step rule reads the least-squares residuals and lands at 0.7616, giving up 0.304 of log-likelihood. Iterating moves it to 0.8080 and gives up 0.002. The maximum is at 0.8044. The curve is not flat between them: what a fixed point of the residual update finds is a solution of a different equation, and the difference is the Jacobian term ½log(1 − ρ²), which grows as the correlation does.

Iterating is not maximising

Re-reading a correlation from the generalised residuals and refitting converges in seven steps. What it converges to solves the first-order condition of a sum of squares, and the likelihood has one term more than that.

together · Dependence 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 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 Two companions on one simulation, two hundredfold apart. How many times as many draws each companion is worth, on the same 4,000 simulated samples of 40 observations. The coverage of the interval is estimated with the observed count as its companion, whose expectation is 12 exactly; they correlate at 0.2665 and the companion is worth 1.08 times the draws. The expected width is estimated with p̂(1 − p̂) as its companion, whose expectation is 0.20475 exactly; they correlate at 0.9977 because the width is a monotone function of it, and the companion is worth 214 times the draws — 856 thousand simulated samples' worth of precision from four thousand.

The check worth more than the check

The same exactly known companion that verifies a simulation can sharpen it. On one set of four thousand draws, one companion is worth 1.08 times the draws and another is worth 214 times them, and the factor is 1 − ρ² with nothing else in it.

method · Routes

Named alongside it

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

Covariate balanceCovariate adjustmentInteractionMarginal distributionMedian splitOrthogonalitySymmetryBasis functionsClosed formParityCopulaExperimental design

All concepts