Extreme values and risk

Adam Butler Adam Butler Biomathematics & Statistics Biomathematics & Statistics

ScotlandScotland

CCTC meeting, September 2007

Extreme values and risk

• Extreme value theory (EVT) is a branch of statistics concerned with the frequency & size of rare events

• EVT methods are widely used in finance, hydrology & engineering, usually for risk assessment, but are not yet widely used in the biological sciences

Extreme values and risk

Risk assessment:

What is the probability we will have more than 100mm of rain on a given day?

Risk management:

I need to build a flood

defense, and I want the

probability that it fails on

any particular day to be

less than 1-in-10000.

How high should it be?

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What is the chance of getting a log

daily return of less than –0.1?

(i.e. a drop in value of 9% or more

since the previous day)

Extreme values and risk

Common features

• We are interested in a process that can be quantified, and for which we have some data

• …and we want to use these data to say something about the probability that a rare or extreme event will occur

Extreme values and risk

• We will usually be interested in events that are beyond the range of the data i.e. we want to extrapolate

• Extrapolation is rarely advisable, but it is sometimes unavoidable, especially when doing risk assessment

• The standard approach would be to assume that the data come from, for example, a normal distribution…

• jk

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P(X < –0.1) 10-20

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…but:

The extreme values don’t play much of a role when we

estimate the parameters, so the model that we end up

fitting might not describe the extreme values at all well…

Empirical: P(X < –0.05) 0.002

Normal: P(X < –0.05) 0.000001

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…and, worse still, extrapolations beyond the range of the

data often differ radically between models that provide

a very similar fit to the bulk of the data…

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Cauchy: P(X < –0.1) 0.02

Normal: P(X < –0.1) 10-20

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In EVT we adopt the principle that we should only make

use of the most extreme data that we have observed

we throw away almost all of the data

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Threshold exceedances

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We consider exceedances of a high threshold

EVT tells us that a good statistical model for exceedances, x, is the Generalised Pareto Distribution (GPD),

P(x) = 1 – [1 + (x / )]-1/ (x > 0)

= “scale parameter”

= “shape parameter”

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GPD: impact of the scale parameter

= 1

= 2

= 3

= 0

= “scale parameter”

= “shape parameter”

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GPD: impact of the shape parameter

= 0

= 1

= -0.5

= 1

= “scale parameter”

= “shape parameter”

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Threshold = u = 25mm

and estimated by maximum likelihood to be 7.70 and 0.108 respectively

P(X > 100) estimated to be 0.0000209

(once in a 131 years)

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…but why is the GPD the “right model” to use?

• In theory: for almost any random variable X, the exceedances of a high threshold u will tend towards the GPD model as u tends towards infinity

• In practice: we use a threshold that is high but still finite: we rely on the fact that if this level is sufficiently high then the asymptotic result will still be approximately true

Extreme value methods

“Parameter stability plot”

Extreme values and risk

Other extreme value models

• A related approach involves analysing the maximum values per day, per month or per year (block maxima)

• EVT suggests that a good model to use in this case is the GEV (Generalised Extreme Value)

;

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Advantages• Robust

Relies on weak assumptions

Avoids bias

• Theoretically soundJustified by asymptotic theory

• Quick & relatively easy to use

• Honest …about the uncertainties involved in making statements about very rare events

Disadvantages

• InefficientMost of the data are thrown away

…we may over-estimate uncertainty

…relies on having a large sample size

• Asymptotics The theory only holds exactly for

infinitely extreme events

Difficult to extend to multivariate case

• Data qualitySensitive to errors in extreme data

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Practicalities

Basic course: http://www.bioss.ac.uk/alarm/training/• Software: routines in… R, Genstat, S-plus, Matlab• Extremes toolkit:

http://www.isse.ucar.edu/extremevalues/evtk.html• Recommended book: Coles (2001) An introduction to statistical

modeling of extreme values. Springer.• Contact me: adam@bioss.ac.uk