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Abstract
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Extreme value theory is still a much underrated part of statistics. In complexity, it compares favourably to classical central limit theory that also depends on the theory or regular variation. For example, the possible limit laws for the maximum of a sample (the extreme value laws), are determined by ¯rst order regular variation and depend on a single parameter, the extreme value index °.
In the case where ° > 0, the statistical estimation of this index is often done by a Hill- type estimator that uses a number of the largest order statistics. It turns out that the bias of such estimators mainly depends on second order properties of the distribution that underlies the sample.In this lecture we give an intuitive and simple overview of first and second order theory.
We show the connection with the above-mentioned bias problem and we indicate how also other problems in extreme value theory can be tackled using this approach. We illustrate the results with a number of examples from environmental sciences and from insurance,especially from catastrophe modeling.
Many results can be found in a textbook J. Beirlant, Y.Goegebeur, J. Segers and J.L.Teugels, Statistics of Extremes, Wiley, 2004.
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