Chapter 5Estimation of the Extreme Value Index
Beirlant J., Herrmann K. and Teugels J.L.
Department of Mathematics and LStat, KU Leuven, Belgium
5.1 Introduction
In Chapter 4, we learned that the stochastic behavior of extremes is well known and can be modeled by the extreme value distribution. The key parameter of this distribution is the extreme value index, or tail index. In this chapter, we deal with the statistical estimation techniques of this parameter. We assume that we have a sample 
of 
independent identically distributed (i.i.d.) or possibly stationary, weakly dependent random variables from an underlying cumulative distribution function 
, and we use the notation 
for the order statistics. Statistical methods of univariate extremes help us to learn from disastrous or almost disastrous events of high relevance in society and with high societal impact. The recent financial crisis has shown that the field of financial statistics certainly fits into this definition. Since the work of Gumbel, summarized in his book in 1958, methodology has been based on the extremal types theorem ...
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