Applications to Modeling Bivariate Extremes

Abstract

The Fisher-Tippet-Gnedenko models for univariate extremes are well understood, as are the corresponding multivariate extensions (see, for example, 89 Galambos (1978, 90 1987) or 140 Resnick (1987)). It should be remembered, however, that the multivariate extreme distributions discussed by these authors correspond to limiting distributions of normalized coordinatewise maxima of sequences of i.i.d. random vectors. Not many multivariate extreme data sets can be reasonably viewed as fitting into this paradigm. For example, if we observe maximum temperatures in the month at several different locations we have a multivariate extreme data set which does not seem to be necessarily explainable in terms of maxima of i.i.d. vectors. For such data sets, some role for univariate extreme distributions seems appropriate but it is not apparent whether it should be a marginal or a conditional role. And of course, even if the data were not generated by a process involving i.i.d. vectors, it might still be well fitted by a multivariate extreme distribution. In many cases, it seems appropriate to approach the problem of modeling multivariate extreme data sets using an augmented toolbox, not just the multivariate extreme models based on maxima of i.i.d. samples. In this chapter we will review some of the popular bivariate extreme models and compare them with certain conditionally specified bivariate extreme models (developed in the spirit of Chapters 4 and 11).