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

Delta method — where it appears

Carrying a bias or variance through a function by multiplying by its derivative, which is a linear approximation and fails where the function curves. Where the function curves sharply over the range the estimate wanders in, the approximation is not merely imprecise but biased in a direction the method cannot see.

Named by 14 essays across 6 fields — each of them below, with the objects they name alongside it.

Where the maximum is, from 15 runs. One dataset, one fitted quadratic, and two answers to "where is the best setting". The delta method reports 0.80 ± 0.46, a finite interval it will report whatever the data does. Fieller's set is 0.49 to 1.76, because the curvature here has t = -4.04. The true optimum is at 0.75.

The optimum is a ratio, and its interval is sometimes the whole line

The best setting is −b₁/2b₂: a ratio of two estimates whose denominator is a curvature the design can often barely see. The delta method reports a finite interval every time and covers 68.8% where the curvature is weak; Fieller's set covers 95% and says so by being unbounded.

surface · Optimum The correction does not arrive at the truth, it passes it. The average decay factor a forecast applies to the last observation, at φ = 0.85 and 50 observations, 3000 series per horizon. The middle curve is φʰ, what the model actually does. Below it is the uncorrected forecast, which uses φ̂ʰ and reverts too fast — 24.8% short at h = 4, 30.0% short at h = 6, 32.7% short at h = 8. Above it is the forecast built on the corrected estimate, which overshoots, and the reason is arithmetic rather than a bad correction: raising an unbiased estimate to a power does not give an unbiased estimate of the power, and the higher the power the more the spread of φ̂ is converted into overshoot.

The repair that moves the wrong number

Correcting the bias in a persistence parameter is one line of arithmetic that works. Feeding the corrected estimate into a forecast repairs the number everybody looks at, makes the forecast worse by squared error at moderate persistence, and improves the interval for a reason that has nothing to do with bias.

evaluation · Bias The interval every package reports first does not cover. Counted coverage of two 95% intervals for the 100-block return level of a normal parent, against the length of the record they were fitted from, over 300 records at each length. The level they are about is known in closed form, so this is coverage of a number rather than agreement between two estimates. The delta-method interval covers 80.3% at 25 blocks and reaches only 89.0% at 200; the profile-likelihood interval sits between 94.0% and 94.7% throughout. The gap is not a small-sample effect that lengthening the record removes — it narrows by 8.7 points for an eightfold longer record.

Two intervals for one return level

Two 95% intervals read off the same fits of the same records, against a level known in closed form. The symmetric one covers 80.3% at twenty-five blocks and reaches only 89.0% at two hundred — and 99.24% of its misses are the interval sitting entirely below the truth, which is not the endpoint anybody expects to fail.

extreme · Extremes The squared estimate 1 standard errors from the flat point, exact and linearised. At δ = √n·μ/σ = 1 the exact law of the squared estimate has mean 2.00, variance 6.00 and skewness 2.177; the delta method's normal has mean 1.00, variance 4.00, no skewness, and 30.85% of its mass below zero, where a square cannot go. The Kolmogorov distance between them is 0.3085.

Where the derivative is zero

The delta method reads a standard error off a tangent line, and at a flat point the tangent says the spread is zero. The interval built on it for a squared mean covers 99.991% there and 85.978% one and a half standard errors away, with nearly every miss on the same side — and the law it should have used is a χ², not a normal.

normal · Clt The average decay factor each route produces, φ = 0.85, 6 steps ahead. The truth is φ^6 = 0.3771. no correction averages 0.2616 with a spread of 0.1646 and a squared forecast error of 3.2516; the formula, on the persistence averages 0.4213 with a spread of 0.2528 and a squared forecast error of 3.4827; the bootstrap, on the persistence averages 0.4355 with a spread of 0.2655 and a squared forecast error of 3.5120; the bootstrap, on the decay factor averages 0.3375 with a spread of 0.2278 and a squared forecast error of 3.4132. 800 series, 100 bootstrap refits each.

Correcting the forecast instead

The complaint against the usual repair is that a correction aimed at the persistence lands on the wrong quantity. Aiming it at the decay factor the forecast actually uses fixes exactly that — the error stops compounding with the horizon, 69.7% becomes 9.5% at twelve steps — and the forecast still gets worse.

evaluation · Bias What the guess is worth, when it is worth anything. The variance cost of an even split relative to the variance-minimising one for a risk difference, against the first arm's proportion, with the second at 0.3. The cost is a pure number: it does not depend on the trial's size. It is exactly zero at 0.3 and at 0.70, where the two arms have the same p(1 − p); it is 0.19% at a half and 4.36% at a tenth. Across the whole range from a tenth to nine tenths it never exceeds 4.36%, which is what the variance-minimising rule is worth here — and what it is worth is the reason it is safe to use with a guess.

The arm whose variance is its answer

With a binary outcome the allocation rule is a function of the proportions the trial exists to estimate. It costs at most 4.36% of variance to ignore it anywhere between a tenth and nine tenths, because √(p(1−p)) stays within a factor of two of its peak across 98% of the unit interval.

allocation · Allocation The two standardised means, and the shape of Fieller's set for their ratio at a = 3, d = 1. Each dot is one pair (zx, zy) drawn around (3, 1). Outside the horizontal band |zy| > 1.96 Fieller's set is a bounded interval, with probability 17.01%; inside the band and outside the disc of radius 1.96 it is everything outside an interval, 75.03%; inside the disc it is the whole line, 7.96%. On 40,000 counted draws Fieller covers ρ = 3.00 95.21% of the time and the delta interval 82.48%.

A ratio whose interval has to be the whole line

The delta interval for a ratio of two means covers 95.61% when the denominator is eight standard errors from zero and 1.10% at a ten-thousandth of one, and ten times as wide it still covers only 3.48%. Linearising is not the fault. Gleser and Hwang proved that every interval that is always finite fails the same way, so an interval that keeps its promise has to be the whole line some of the time.

normal · Clt Four 95% intervals for the odds after 2 of 20. credible, transformed: 0.0218 to 0.3964. Wald, transformed: -0.0305 to 0.3012. delta method on the odds: -0.0512 to 0.2734. delta method on the log-odds: 0.0258 to 0.4789. The first two are the same intervals for the proportion with their endpoints put through the odds; the last two are fresh approximations made on the new scale.

An interval for something else

An interval for the odds is free — put the endpoints through the odds and the coverage does not move, exactly, for any interval at all. The method everyone uses instead computes a new standard error on the new scale, and at twenty trials that costs four points of coverage, produces negative odds, and has no value at all when nothing was observed.

bayes · Credible Three contrasts on one dataset, three different splits. The variance-minimising allocation for each of three ways of reporting the same two-arm comparison, against the first arm's proportion, with the second at 0.1. A risk difference wants the arm with the larger p(1 − p) to get more units; a log odds ratio wants it to get fewer, and the two curves are exact reflections of each other in the half line. A log risk ratio wants something else again. At a first-arm proportion of 0.6 they ask for 62.0%, 21.4% and 38.0% of the units. A trial reporting more than one of them cannot be optimal for either.

Two contrasts, one split

A risk difference wants 62.0% of the units in the first arm, a log risk ratio wants 21.4% and a log odds ratio wants 38.0% — on one dataset, with one pair of proportions. The difference's rule and the odds ratio's are exact reflections of each other, so no split can be near-optimal for both.

allocation · Allocation What the forecast interval is short by, φ = 0.85, 6 steps ahead. The plug-in interval covers 88.42% against a claimed 95%. Correcting the variance recovers 0.56 points, propagating the persistence's own standard error recovers 0.40, correcting the persistence recovers 2.66, and all three together recover 4.20 — leaving 2.38 points unaccounted for.

What the interval is short by

The forecast interval covers 88.42% where it claims 95%. Correcting the persistence recovers 2.66 points, correcting the innovation variance 0.56, propagating the persistence's own standard error 0.40 — and all three together recover 4.20 of the 6.58, leaving a residual none of the standard repairs reaches.

evaluation · Bias The law is the eigenvalues, and nothing else. The mean and the skewness of n(ĝ − g) at the stationary point, measured over 40,000 draws, against the closed forms: half the sum of the Hessian's eigenvalues for the mean, and for the skewness twice root two times the sum of their cubes over the sum of their squares to the power three halves. The worst disagreement anywhere is 0.028. In one variable the second-order law is a single chi-square with one degree of freedom and its sign is the sign of the second derivative; here it is a weighted sum with the Hessian's eigenvalues as weights, so a bowl and a valley differ in both moments and a saddle has both equal to zero.

A flat point with more than one direction

At a stationary point of a function of several means the second-order law is ½ Z′HZ, so the bias is half the Hessian's trace — 2.008 for a bowl, 5.028 for a valley, and −0.006 for a saddle, where the eigenvalues cancel. The saddle's coverage is the worst of the three.

normal · Clt Five 95% intervals for an AR(1) forecast 6 steps ahead at φ = 0.85, fifty observations. Coverage on 4,000 series, the same series for every interval, each simulated interval built from 499 simulated futures. The plug-in formula: 88.48%. The formula, all three repairs: 92.80%. Simulated: parameters drawn: 92.55%. Simulated: residual bootstrap: 88.25%. Simulated: bootstrap, bias removed: 93.47%.

An interval built from simulated futures

A forecast interval read off simulated futures needs no delta method and no normal multiplier, and was expected to close the residual the three repairs to the formula leave. It does not close it on its own: the plain residual bootstrap covers 88.25% six steps ahead at φ = 0.85, the plug-in formula's 88.48% to within the count, because it simulates around the same biased persistence. With that bias removed inside the bootstrap it covers 93.47%, the best of five intervals, and its shortfall stops growing with the horizon — 93.80% at twelve steps, where the repaired formula covers 91.85%.

evaluation · Bias Testing for flatness at 5% before choosing the interval for a squared mean. Coverage of four nominal 95% intervals for μ², exact, against the true mean's distance δ from the flat point in standard errors. The tangent interval Z² ± 2z|Z| has its worst, 85.98%, at δ = 1.45. The second-order interval read off χ²₁ is right only at δ = 0 and falls to 34.22% at δ = 2. The two-stage rule, which uses the second-order interval when a 5% test does not reject flatness, has its worst, 49.72%, at δ = 1.96. The squared exact interval never falls below 95%.

Testing for a flat point first

Test whether the gradient is zero, and if the test does not reject, read the interval off the second-order law that holds at a stationary point. For a squared mean that two-stage rule covers about 50% at its worst whatever the pretest's level — 49.72% with a 5% test, where the tangent interval it was meant to rescue never falls below 85.98% — because its worst sits exactly where the true distance equals the test's critical value, half the samples are declared flat there, and the second-order interval cannot reach the truth from any of them.

normal · Clt How much of a return level's uncertainty is the extremal index's, by how far past the record the level is. Delta-method variance of the level from a declustered peaks-over-threshold fit, averaged over 300 records of 4,000 observations. The cluster rate's share at 1.5, 2, 5, 20, 100, 1000 periods: independent, 0.374, 0.253, 0.084, 0.027, 0.012, 0.005; runs of exceedances, θ = 0.6, 0.558, 0.396, 0.141, 0.039, 0.015, 0.006; spaced clusters, θ = 0.4, 0.845, 0.562, 0.222, 0.057, 0.019, 0.007; runs of exceedances, θ = 0.25, 0.787, 0.788, 0.327, 0.083, 0.024, 0.008.

Where the extremal index matters to a return level

A return level from a declustered fit depends on how often clusters arrive, which is the extremal index estimated from the same record. For a level reached once in two periods of a twenty-period record the index's error is a quarter to four fifths of the level's variance, and an interval that leaves it out covers 59.0% when clusters are long; put in, 90.0%. For a level reached once in a hundred periods its share is under two and a half per cent and adding it changes no interval at all — what fails there is the shape, fitted from however many clusters the dependence left: 82.3% coverage from forty-six clusters, 65.7% from nineteen.

extreme · Extremes

Named alongside it

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

CoverageConfidence intervalMonte CarloClosed formPlug in estimateBias correctionForecast horizonStationarityStationary pointBinomial proportionBootstrapChi-square

All concepts