Concept

Declustering — where it appears

Grouping the exceedances of a threshold into clusters and keeping one value from each, so that what is fitted behaves like independent extremes. The rule for where a cluster ends decides how many clusters there are, and a fixed run length splits every cluster whose members are not neighbours.

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

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

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.

extreme · Extremes

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.

extreme · Extremes

Named alongside it

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

Closed form Cluster size Exceedance Extremal index Fréchet law Generalised Pareto Independence Peaks over threshold Return level Shape parameter Standard error Autocorrelation

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