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

Upper endpoint — where it appears

The largest value a distribution puts any probability below, finite only when the extreme-value shape is negative. A fitted shape below zero is therefore an assertion that the quantity has a ceiling, estimated from a record that has not reached it.

Named by 3 essays across one field — 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 The threshold buys accuracy and spends exceedances. The mean squared error of the estimated shape against the threshold, split into the square of its bias and its spread, over 600 records of 2000 readings from a a normal parent. At the 0.9 quantile 199 exceedances are left, the bias is -0.1708, the spread is 0.0701 and the total error is 0.0341. The bias falls as the threshold rises because the exceedances get closer to being generalised Pareto; the spread rises because there are fewer of them. The sum is smallest at the 0.925 quantile, at 0.0340, of which 80.6% is still bias — so even the best threshold on this grid is one where accuracy, not spread, is the binding constraint.

The threshold is a dial

A peaks-over-threshold analysis has one knob, and raising it buys accuracy with exceedances. For a normal parent the error is smallest at the 0.925 quantile and 80.6% of it is still bias there — and both diagnostics practitioners use to set the knob lose to a fixed 0.90 rule, one by a factor of 1.590 and one by 11.881.

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.

Shape parameterBlock maximaBlock sizeClosed formEstimation errorExtreme value theoryGeneralised extreme valueGumbel lawMaximum likelihoodMean squared errorSampling distributionTail index

All concepts