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

Convolution — where it appears

The distribution of a sum of two independent quantities, built by combining their distributions over every way the sum can arise. It predicts what adding two statistics does to a tail, which is why the critical value of a sum sits below the sum of the two critical values.

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

An overlap that depends on where it is read. Each pair's overlap on the doubled log-likelihood scale — the sum of the two searches' own quantiles less the joint search's quantile, as a share of the smaller — at five quantile levels, over 1200 draws under the first-order autoregression. two dictionaries of independent columns: -0.176 at 0.5, -0.000 at 0.75, 0.238 at 0.9, 0.320 at 0.95, 0.538 at 0.99; a break, and an independent column: -1.211 at 0.5, -0.406 at 0.75, -0.029 at 0.9, 0.205 at 0.95, 0.161 at 0.99; two dictionaries of step columns: -0.397 at 0.5, 0.145 at 0.75, 0.438 at 0.9, 0.541 at 0.95, 0.682 at 0.99; a break, and a step column it contains: 1.000 at 0.5, 1.000 at 0.75, 1.000 at 0.9, 1.000 at 0.95, 1.000 at 0.99; a break, and a whitening window: 0.512 at 0.5, 0.640 at 0.75, 0.716 at 0.9, 0.732 at 0.95, 0.754 at 0.99. Every pair but the containment overlaps more at the 95% point than at the median, the control included.

The overlap at the critical value

Two searches over independent noise columns are additive on average: the joint search finds what the two find separately, to within a fortieth of the smaller. At the point a test actually uses — the joint statistic's 95% quantile — the same pair overlaps by 0.320, and a rule that adds the two searches' own critical values rejects 1.4% of true nulls instead of 5%. Nothing about the searches causes it. Quantiles of a sum do not add, and on every pair in the field the overlap at the critical value is larger than the overlap the field measured.

apart · Criterion The overlap climbs and the interaction does not. The over-charge at each quantile level split into overlap — the two searches' own quantiles less the sequential run's — and interaction — the joint search's quantile less the sequential run's — for three pairs over 1200 draws. two independent dictionaries: overlap -0.40, 0.01, 1.55, 2.44, 5.72; interaction 0.07, 0.01, 0.11, 0.06, 0.00. two step dictionaries: overlap -0.73, 8.19, 22.53, 31.02, 47.75; interaction 4.87, 4.55, 4.90, 4.75, 1.10. a break and a window: overlap 17.88, 29.52, 43.01, 53.79, 71.61; interaction 1.57, 1.29, 1.19, 3.22, 2.75, at the 50%, 75%, 90%, 95% and 99% points.

The split at the critical value

Split the over-charge of two searches at the joint statistic's 95% point rather than at its mean, and the two components behave differently. The interaction — what only the joint search can reach — stays within two units of its average at every level up to the 95% point. The overlap climbs from nothing to most of the over-charge, on two searches over independent columns as much as on searches that share ground. At the critical value the split's overlap has stopped meaning shared ground: it is mostly the arithmetic of adding tails, and the identity that defines it hides that.

separate · Break point

Named alongside it

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

Change pointCombinatorial searchCritical valueData snoopingLikelihood ratioMonte CarloQuantileCalibrationModel selectionSize of a testSpecification searchSupremum statistic

All concepts