Extreme Value Applications and Theory

Extreme Value Applications

Understanding the distribution of extreme events has a number of practical applications. For example:

  • Based on the annual highest tide measurements over the past 20 years, determine how high a barrier needs to be to avoid a once-in-a-century catastrophe where water overflows the barrier and floods the community.
  • Other extreme events that can be studied: extreme temperature (cold or heat), wind speed (e.g. in a hurricane), water (drought or floods),  earthquakes, forest fires or financial collapse.

These types of applications can be addressed using the Generalized Extreme Value (GEV) distribution.

Extreme Value Theory

Just as the Central Limit Theorem states that for a sufficiently large sample from (almost) any distribution, the sample mean will be normally distributed, the Extreme Value Theorem (aka the Fisher-Tippett-Gnedenko Theorem) states that for a certain class of distributions, the maximum value for a sufficiently large sample will have a GEV distribution.

More precisely, the maximum value of a sample, denoted x(n) where n is the size of the sample, has a GEV distribution for n sufficiently large, i.e. there are parameters μn, σn, ξn such that x(n) → GEV(μn, σn, ξn) as n → ∞.

If a sample comes from a beta distribution (including the uniform distribution) then the maximum value (for a sufficiently large sample) has a reverse Weibull distribution. If the sample comes from a Pareto, Fréchet or t-distribution, then the maximum value has a Fréchet distribution. Finally, if the sample comes from a Weibull, exponential, gamma, logistic, normal or log-normal distribution then the maximum value has a Gumbel distribution.

Properties

Property 1: For constants, a and b, with b ≠ 0, if x ∼ GEV(μ, σ, ξ), then bx + a ∼ GEV(bμ+a, bσ, ξ).

Thus, if z = (x–a)/b  then z ∼ GEV(μ, σ, ξ), then x = a + bz ∼ GEV(bμ+a, bσ, ξ).

Property 2: If x1, …, xn ∼ Exp(1), then x(n) ∼ GEV(ln n, 1, 0) as n → ∞.

Proof: The cdf of Exp(1) is F(x) = 1 – e–x

Thus

Proof 1

Proof 2

As stated in Built-in Excel Functions

Exp(x) formula

Thus, as n → ∞

Proof 3

which means that in the limit

Property 4

By Property 1, it follows that as n → ∞

Proof 5

Property 3: For x1, …, xn ∼ Frechet(α, β, γ) then

Distribution of sample maximum

Proof: We will prove the case where α = 1, β = 1, γ = 0. The cdf in this case is  F(x) = e-1/x. Thus

Proof 1

 

Proof 2

By Property 1 (Frechet version), it follows that

Proof 3

References

Fawcett, L. (2012) Classical model for extremes
http://www.mas.ncl.ac.uk/~nlf8/teaching/mas8391/background/chapter2.pdf

Charras-Garrido, M. and Lezaud, P. (2013) Extreme value analysis: an introduction. Journal de la Société Française de Statistique
https://www.semanticscholar.org/paper/Extreme-Value-Analysis%3A-an-Introduction-Charras-Garrido-Lezaud/de1b2faefe7ea970f9d29a8bf6925cd83e01b4c1

Friederichs , P. (2007) An introduction to extreme value theory
https://www.scribd.com/document/315970452/evt-cops2

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