Concept

Second-order delta method — where it appears

The expansion used where a function's gradient vanishes, so that the first-order term contributes nothing. The limit is a weighted sum of squared normals with the Hessian's eigenvalues as weights, which is skewed when they agree in sign and symmetric when they cancel.

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

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 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

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

Named alongside it

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

Chi-square Closed form Confidence interval Coverage Delta method Stationary point Central limit theorem Eigenvalue Monte Carlo Non-central χ² Saddle point Skewness

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