Contents 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 7.2 Classical results on componentwise maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . 374

7.2.1 Univariate extreme events: Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 7.2.2 Multivariate extreme value distributions . . . . . . . . . . . . . . . . . . . . . . . . 375

7.3 An alternative modeling approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 7.4 Measuring extremal dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

7.1 Introduction

A wide variety of situations concerned with extreme events has an inherent multivariate character, as pointed out by Coles and Tawn (1991). Let us consider, for example, the oceanographic context, and focus on the sea-level process. Such a variable can be divided into several physical components like mean-level, tide, surge, and wave, which are driven by different physical phenomena (see, for example, Tawn (1992) for details). Moreover, extreme sea conditions leading to damages are usually a consequence of extreme values jointly in several components. The joint structure of the processes has therefore to be studied. Another type of dependence that can be of great interest is the temporal one: high sea levels can be all the more dangerous when they last for a long period of time. Therefore, a given variable observed at successive times is likely to contain crucial information. Other examples of applications have been listed recently by Kotz and Nadarajah (2000), concerning, among others, pollutant concentrations (Joe, Smith, and Weissman, 1992), reservoir safety (Anderson and Nadarajah, 1993), or Dutch sea dikes safety (Bruun and Tawn, 1998; de Haan and de Ronde, 1998).