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The asymptotic theory of sample extremes has been developed in parallel with the central limit theory, and in fact the two theories bear some resemblance.

The extreme value condition for each , with 1 + > 0, or equivalently for each > 0, where is a real constant called the , is designed to allow convergence in distribution of normalized sample maxima, as in (1.1.1). But the conditions also imply convergence of other high-order statistics.

The alternative conditions of Theorem 1.1.6 (Section 1.1.3) serve as a basis for statistical applications of extreme value theory.

With the sea level case study introduced in Section 1.1.4 and further discussed in Section 3.1, we illustrated the role of extreme value theory in extreme quantile estimation. In the sequel we explore this example a bit further.

Chapters 1–4 constitute the basic probabilistic and statistical theory of one-dimensional extremes. In this chapter we shall present additional material that can be skipped at first reading. It is not used in the rest of the book.

In order to see the usefulness of developing multivariate extremes we start with a problem for which multivariate extreme value theory seems to be relevant.

In Chapter 6 we have seen that a multivariate extreme value distribution is characterized by the marginal extreme value indices plus a homogeneous exponent measure or alternatively a spectral measure. In particular, there is no finite parametrization for extreme value distributions. This suggests the use of nonparametric methods for estimating the dependence structure, and in fact we are going to emphasize those methods.

In this chapter we are going to deal with methods to solve the problem posed in a graphical way in Chapter 6. The wave height (HmO) and still water level (SWL) have been recorded during 828 storm events that are relevant for the Pettemer Zeewering. Engineers of RIKZ (Institute for Coastal and Marine Management) have determined , that is, those combinations of HmO and SWL that result in overtopping the seawall, thus creating a dangerous situation. The set of those combinations forms a failure set .

Infinite-dimensional extreme value theory is not just a theoretical extension of the theory to a more abstract context. It serves to solve concrete problems. We start with a motivating example.

In Section 9.1 we considered the following mathematical problem: given independent and identically distributed random functions , ,..., in [0, 1] whose distribution is in the domain of attraction of an extreme value distribution in [0, 1], estimate the probability (X(s) > (s), for some s ∈[0,1], where is a given continuous function.