Extreme value theory

Two main approaches exist for practical extreme value analysis.

The first method relies on deriving block maxima (minima) series as a preliminary step.

[1] For AMS data, the analysis may partly rely on the results of the Fisher–Tippett–Gnedenko theorem, leading to the generalized extreme value distribution being selected for fitting.

[2][3] However, in practice, various procedures are applied to select between a wider range of distributions.

Given that the number of relevant random events within a year may be rather limited, it is unsurprising that analyses of observed AMS data often lead to distributions other than the generalized extreme value distribution (GEVD) being selected.

[4] For POT data, the analysis may involve fitting two distributions: One for the number of events in a time period considered and a second for the size of the exceedances.

With the help of R.A. Fisher, Tippet obtained three asymptotic limits describing the distributions of extremes assuming independent variables.

One universality class of particular interest is that of log-correlated fields, where the correlations decay logarithmically with the distance.

Extreme value theory in more than one variable introduces additional issues that have to be addressed.

it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations.

In the multivariate case, the model not only contains unknown parameters, but also a function whose exact form is not prescribed by the theory.

[27][28] It is not straightforward to devise estimators that obey such constraints though some have been recently constructed.

[29][30][31] As an example of an application, bivariate extreme value theory has been applied to ocean research.