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

Return period — where it appears

The average number of periods between exceedances of a level: the T-period level is passed once in T periods on average. It fixes how far past a record a return level reaches, and so how much of its uncertainty is the fitted shape's rather than the data's.

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

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 How much of a return level's uncertainty is the extremal index's, by how far past the record the level is. Delta-method variance of the level from a declustered peaks-over-threshold fit, averaged over 300 records of 4,000 observations. The cluster rate's share at 1.5, 2, 5, 20, 100, 1000 periods: independent, 0.374, 0.253, 0.084, 0.027, 0.012, 0.005; runs of exceedances, θ = 0.6, 0.558, 0.396, 0.141, 0.039, 0.015, 0.006; spaced clusters, θ = 0.4, 0.845, 0.562, 0.222, 0.057, 0.019, 0.007; runs of exceedances, θ = 0.25, 0.787, 0.788, 0.327, 0.083, 0.024, 0.008.

Where the extremal index matters to a return level

A return level from a declustered fit depends on how often clusters arrive, which is the extremal index estimated from the same record. For a level reached once in two periods of a twenty-period record the index's error is a quarter to four fifths of the level's variance, and an interval that leaves it out covers 59.0% when clusters are long; put in, 90.0%. For a level reached once in a hundred periods its share is under two and a half per cent and adding it changes no interval at all — what fails there is the shape, fitted from however many clusters the dependence left: 82.3% coverage from forty-six clusters, 65.7% from nineteen.

extreme · Extremes

Named alongside it

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

Return levelBlock maximaBlock sizeClosed formConfidence intervalCoverageDeclusteringDelta methodEffective sample sizeEstimation errorExtrapolationExtremal index

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