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

Block size — where it appears

How many readings go into each maximum a block-maxima analysis fits. It decides how close those maxima are to their limit law, so a bias in the fitted shape falls with the block size and not with the number of blocks.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

More blocks, and the light tail is called wrong more often. The share of records whose three-way family call is right, over 400 records at each block count, at 100 readings a block. A 95% interval for the shape is formed and the record is recorded as calling Fréchet, Weibull or Gumbel according to whether that interval sits above zero, below it, or straddles it. The two signed parents go from 51.5% and 71.0% at 20 blocks to certainty by 100. The light-tailed one goes the other way — 83.0%, 75.5%, 55.0%, 36.5%, 4.8% — because its estimate sits at about -0.1157 whatever the record length, and a longer record only shrinks the interval onto that number.

Three shapes, one limit

A normalised sum has one limit and a normalised maximum has three, indexed by a single number. Twenty blocks put the sign of that number right 97.3% of the time — and naming the family from a light-tailed record gets worse as the record grows, from 83.0% at twenty blocks to 4.8% at five hundred.

extreme · Extremes One tail arrives; the other is still on its way at a million. The Kolmogorov distance between the exact law of a normalised maximum and its Gumbel limit, at six block sizes, for two parents that both have that same limit. Both are closed form: the exact law of a maximum is F(x)^n and no simulation is involved. The exponential parent's distance falls from 0.0280 to 2.707e-7 — a factor of a hundred thousand, which is exactly one over n. The normal parent's falls from 0.0522 only to 0.0091, a factor of 5.74, because its rate is one over log n. At a million readings a block the two differ by a factor of 33556.3.

The maximum converges slowly

The rate at which a normalised maximum reaches its limit law is computable rather than simulable, because the exact law of a maximum is always available. For a normal parent the distance falls like one over the logarithm of the block and is still 0.0091 at a million readings; for an exponential parent, with the same limit, it is 2.707×10⁻⁷.

extreme · Extremes The record stops here, and the curve does not. The level exceeded once in T blocks, against T, for a normal parent at 365 readings a block. The truth is closed form — the block maximum's own distribution function is Φ(x) raised to the 365, so the T-block level is Φ⁻¹ of the 365-th root of (1 − 1/T), with nothing fitted in it — and the fitted mean over 800 records of 50 blocks sits on top of it, 4.0186 against 4.0330 at 100 blocks. What moves is not the level but its error, which grows from 0.0956 at 10 blocks to 0.6383 at a thousand while the level itself moves only from 3.4421 to 4.5454. The rule marks the largest reading an average record contains, 4.0062: everything to the right of where it crosses is read from a fit rather than from data.

A level with no data in it

The largest of fifty block maxima is a 51-block event by its own plotting position, so a hundred-block level is read 1.96 times past the longest event the record contains — and it lands above the largest reading on 52.4% of records. The estimate stays nearly unbiased out there; what grows is its error, sixfold from ten blocks to a thousand.

extreme · Extremes Coverage as the effect moves with the level. Coverage on ten and two in every block, with the level between blocks moving at a spread of 0.5 and the effect moving with it at each slope, 4000 runs each. At -3: refusing the sums 94.4%, pooling them 88.8%, residualised 97.0%; At -2.5: refusing the sums 94.3%, pooling them 88.9%, residualised 95.6%; At -2: refusing the sums 94.9%, pooling them 90.0%, residualised 93.8%; At -1.5: refusing the sums 95.1%, pooling them 91.7%, residualised 92.7%; At -1: refusing the sums 95.1%, pooling them 94.8%, residualised 94.0%; At -0.5: refusing the sums 95.1%, pooling them 97.2%, residualised 97.7%; At 0: refusing the sums 95.0%, pooling them 98.8%, residualised 99.6%; At 0.5: refusing the sums 95.0%, pooling them 99.4%, residualised 99.9%; At 1: refusing the sums 95.0%, pooling them 99.4%, residualised 99.9%.

The sums under a moving level

A trial's block sums are a free second estimate of its variance only while they see what the differences see, and a level that moves between blocks with an effect moving against it breaks that: pooled raw at a ceiling they cover 87.6% to 90.0%. Residualising the sums on an unbalanced design does not make them safe. It moves the failure. On ten and two, 90.0% at the ceiling becomes 93.8%, the trough leaves for a slope of about −1.5, and over the rest of the range the same estimate over-covers, at 99.9% on an interval half as wide again as it needs to be.

contrast · Blocking Coverage lost against the honest rule. Coverage lost in points against the rule that reads only within-block spreads, on the same observations, blocks of 2, 6000 runs, at half-widths 0.5, 0.35, 0.25, 0.15. the reported spread in full: 0.32 ± 0.27, 2.53 ± 0.37, 5.33 ± 0.41, 3.75 ± 0.35. a fixed half: 0.72 ± 0.26, 2.43 ± 0.29, 0.98 ± 0.26, -0.25 ± 0.20. κ = (b − 1)/(N − 1): 0.37 ± 0.21, 1.18 ± 0.25, 0.58 ± 0.23, -0.18 ± 0.19. the same, phased in as b/(b + 20): 0.22 ± 0.10, 0.35 ± 0.16, 0.23 ± 0.18, -0.10 ± 0.19. the same, read in only after twenty blocks: 0.00 ± 0.00, -0.02 ± 0.04, 0.17 ± 0.19, -0.17 ± 0.19.

A rule that sets its own κ

A stopping rule may read the spread inside its blocks and may not read the spread between them, and the dial between the two costs nothing for a long stretch and then a lot. The share an efficiency argument recommends is each estimate's share of the degrees of freedom, and on blocks of two at a half-width of 0.15 that loses nothing and steadies the experiment by 30%. At 0.35 it loses 1.18 points. Deferring the forbidden reading until twenty blocks costs nothing measurable at any requirement, and it keeps most of the steadiness where steadiness can be had.

pace · Stopping

Named alongside it

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

Block maximaClosed formExtreme value theoryGeneralised extreme valueGumbel lawShape parameterCentral limit theoremCoverageDegrees of freedomDomain of attractionFixed-width intervalMonte Carlo

All concepts