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

Fieller interval — where it appears

The exact confidence set for a ratio of two normal means, found by testing every candidate ratio with a statistic that is linear in the data. It is a bounded interval, the outside of an interval or the whole line, and it covers its stated level because it is allowed to be unbounded.

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

Also named here as ratio estimator — the same set of essays touches all of them, so they are one junction rather than several.

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 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

Named alongside it

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

Confidence intervalCoverageDelta methodRatio estimatorBootstrapCentral composite designConditional coverageCurvatureGleser–Hwang theoremMonte CarloNon-central χ²Percentile interval

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