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

Stationary point — where it appears

A point where a function's first derivative is zero, so its tangent line is flat. There the delta method gives an estimate no spread at all, and the real law comes from the second derivative: a scaled χ² at rate n rather than a normal at rate √n.

Named by 6 essays across 2 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 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 What the fit calls the shape, against what it is. One eigenvalue held at −3 and the other swept from −2 to 2, so the truth is a maximum on the left and a saddle on the right and the change happens at exactly zero. At an eigenvalue of −0.25 — a genuine maximum — the fit reports a saddle on 26.4% of studies; at +0.25 — a genuine saddle — it reports a maximum on 25.1%. The standard error of a squared coefficient under this design is 0.3791, and the region of confusion is about that wide either side of zero.

The sign the curvature has

A fitted surface reports a maximum, a minimum or a saddle, and the report is a comparison of two estimated eigenvalues against zero. At a true second eigenvalue of −0.25 the fit calls a genuine maximum a saddle on 26.4% of studies, and at +0.25 it calls a genuine saddle a maximum on 25.1%.

surface · Optimum The ridge, when the fitted optimum is outside the region. One fitted surface. Its stationary point is at a radius of 2.289 and the fit calls the shape a maximum. The ridge is the best setting at each radius, found by the Lagrange condition (B̂ − μI)x = −ĝ/2; the fitted response rises along it from 59.93 at the centre to 62.10 at the edge. The true optimum is at (0.4, 0.3).

When the best setting is outside the region

On a flat surface at twice the noise the fitted optimum lands outside the experimental region on 24.9% of studies and more than three coded units out on 11.8%. The answer is a ridge — the best setting at each radius, with a closed form — and the two obvious rules for using it turn out to be within four per cent of each other.

surface · Optimum What a confirmation run at the chosen setting would find. The true optimum is worth 62.348. At σ = 2 the fit predicts 62.679 at the setting it recommends and the truth there is 61.821 — a gap of 0.858, which is 0.72 of the prediction's own standard error. The setting itself gives up 0.527 against the best available.

The run that confirms it

The setting a response-surface analysis recommends was chosen because the fitted surface was highest there, so the height the fit predicts at it is a maximum over a random field. At twice the noise the fit predicts 0.858 more than is there — 0.72 of the prediction's own standard error — and the gap is not noise, it is the selection.

surface · Optimum 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 ½ Σλ and 2√2 Σλ³ ⁄ (Σλ²)^(3⁄2). The worst disagreement anywhere is 0.028. In one variable the second-order law is a single χ² and its sign is the sign of g″; 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

Named alongside it

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

Saddle pointCentral composite designConfidence intervalCoverageCurvatureDelta methodEigenvalueExperimental designMonte CarloResponse-surfaceCanonical analysisCentral limit theorem

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