Extremes — the series · Normal Distribution

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.

part 1 · extreme

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.

part 2 · extreme

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.

part 3 · extreme

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.

part 4 · extreme

Two intervals for one return level

Two 95% intervals read off the same fits of the same records, against a level known in closed form. The symmetric one covers 80.3% at twenty-five blocks and reaches only 89.0% at two hundred — and 99.24% of its misses are the interval sitting entirely below the truth, which is not the endpoint anybody expects to fail.

part 5 · extreme

The exceedances arrive together. 300 steps of a max-autoregression with dependence 0.75, drawn on a logarithmic scale because its marginal has no variance. The rule marks the 0.9 quantile: 30 of the 300 readings are above it and they fall into 5 clusters, the largest holding 11. The mean cluster holds 6.000, and its reciprocal — 0.167 — is the runs estimator of the extremal index, whose true value for this process is exactly 1 − 0.75 = 0.25. Every threshold method in the collection assumes exceedances are independent pieces of information; here 30 of them are 5.

The clustering the tail has

Every threshold method counts exceedances as though they were independent pieces of information, and in a dependent series they arrive in clusters. Ignoring that overstates a return level by the reciprocal of the extremal index — ×3.527 counted where the mean cluster holds four — and leaves a reported standard error 2.151 times too small.

part 6 · extreme

A run length of 4 makes every exceedance its own cluster. 200 steps of a max-moving-maximum, X(t) = max(0.4·Z(t), 0.3·Z(t−6), 0.3·Z(t−12)) with unit Fréchet innovations Z, whose extremal index is exactly 0.40: one large innovation can put three readings above a threshold, 6 steps apart. The rule marks the 0.9 quantile and 19 readings clear it; 12 of the gaps between consecutive exceedances are exactly 6 steps. With a run length of 4, so that two exceedances 4 or more steps apart start separate clusters, they form 19 clusters, shaded, the largest holding 1. The runs estimator reads 1.000 against 0.40.

The run length a declustering chooses

The runs estimator of an extremal index carries a constant nobody derives. Where a cluster is a run of neighbouring exceedances the constant barely matters; where a cluster's members fall six steps apart, the estimate is 0.9069 at a run length of six and 0.3649 at seven against an index of 0.40, and a run length of four removes under a tenth of the overstatement declustering exists to remove. A rule that reads the run length off the data has the smallest worst error of the three.

part 7 · extreme