Module 20: Extreme Value Theory

Question 1

Why might the form of a distribution be incorrect in the tails? (3)

Answer 1

• the ‘true’ distribution is more skewed or leptokurtic than is indicated by the available data.
• the parameter estimates are inappropriately influenced by the main bulk of the data in the middle of the distribution
• features change over time, eg heteroscedasticity, structural breaks.

Question 2

GEV distribution

Answer 2

If:

• losses Xᵢ are iid with cumulative distribution F
• Xₘ = max( X₁, X₂, …, Xₙ ) are the lock maxima
• β₁, β₂, …, βₙ > 0 and α₁, α₂, …, αₙ are a suitable sequence of real constants

then, if n is sufficiently large, the distribution of the standardised block maxima (Xₘ - αₙ) / βₙ is approximately described by the Generalised Extreme Value (GEV) family of distributions.

Question 3

3 parameters of the GEV family

Answer 3

• location (α)
• scale (β)
• shape (γ)

Question 4

The GEV distribution can be used to analyse a set of observed losses in two different ways

Answer 4

1. select the maximum observation in each block (the return-level approach)
2. count the observations in each block that exceed some set level (the return-period approach)

Question 5

GPD Distribution

Answer 5

Let X be a random variable with the distribution function F.

If the losses are independent and identically distributed (iid) then, as the threshold increases, the distribution of the conditional losses (the exceedences) will converge (whatever the underlying distribution of the data) to a Generalised Pareto distribution.

Question 6

2 Parameters of the GPD family

Answer 6

• scale (β)

- shape (γ)

Question 7

Mean excess function, e(u)

Answer 7

e(u) = E( X - u | X > u )

Question 8

How to select a suitable threshold (above which a GPD is fitted to the data)

Answer 8

Determine the lowest threshold above which the mean excess function, e(u) = E( X - u | X > u), is linear in u.

The slope of this line is related to the shape parameter, γ. Typically, the chosen threshold is likely to be around the 90-95th percentile of the complete distribution.

Above this distribution, a GPD can be fitted to the selected data by using MLE or the method of moments to determine the parameters.

There is a trade-off between the quality of approximation to the GPD (good for high thresholds) and level of bias in the fit (lower for lower thresholds).

Question 9

Asymptotic property

Answer 9

When the maxima of a distribution converge to a GEV distribution (which is the case for all commonly used statistical distributions), the excess distribution converges to a GPD distribution with an equivalent shape parameter, γ.