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

Extreme value theory — where it appears

The limit theory for the largest value in a sample rather than for its average. It says that a normalised maximum, if it converges at all, converges to one of exactly three laws, indexed by a single shape parameter — so an analysis of extremes is a claim about which of the three a record came from.

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

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

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

Named alongside it

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

Block maximaBlock sizeGeneralised extreme valueGumbel lawShape parameterCentral limit theoremClosed formDomain of attractionOrder statisticReturn levelSampling distributionUpper endpoint

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