Concept

Berry–Esseen — where it appears

A theorem bounding how far a standardised sum's distribution function can be from the normal at any sample size. The bound falls as one over the root of n and is set by the source's third absolute moment, and it is nearly attained by a coin and loose by a large factor on smooth sources.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The normal approximation's error on a sum of 100 exponential draws, under its Berry–Esseen bound. The distance between the exact distribution function and the normal one peaks at 0.0133, at z = -0.01. The Berry–Esseen bound is 0.1146, 8.62 times the real worst error, and larger than the whole 2.5% tail a two-sided test reads.

A bound written for a coin

The Berry–Esseen theorem guarantees how far a standardised sum can be from the normal, and the guarantee is true. On an exponential source it is 8.62 times the real worst error at every sample size, the worst error sits at the centre rather than in a tail, and at a hundred draws the bound is larger than the 2.5% tail it would be asked to vouch for.

expansion · Rate

Where the normal approximation converges, and where it does not. Relative error against the exact binomial. At n = 1280 the error at the median is 0.96% and three sigma out it is 25.7% — a factor of 27. The tail is where the approximation is used.

The tail converges last

The central limit theorem is usually shown as a shape arriving. What the demonstration leaves out is the rate — and the rate is wildly different in the middle and in the tail, which is where every approximation in the subject is actually read.

normal · Rate

Named alongside it

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

Central limit theorem Convergence rate Normal approximation Skewness Tail probability Discreteness p-value Sample size Standard deviation Worst case

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