Ch. 17. Extreme value theory, models and simulation

Publisher Summary Extreme value theory concerns the behavior of the extremes of a process or processes. The fundamentals of this probabilistic theory have been known since the beginning of the twentieth century. But the relevant statistical models emerged only much more recently. The aim of this chapter is to review the fundamentals, relevant statistical models and simulation schemes, and to note the various applications that the models have attracted. Most of the material presented is for independent and identically distributed observations. This chapter discusses the development of extreme value theory for nontrivial stochastic processes. Univariate extremes are traditionally modeled by the annual maximum method. The theoretical motivation for it comes from the well known extremal types theorem: details of this limit theorem, its variants and corresponding rate of convergence results. A major weakness of the annual maximum method is that it utilizes only the largest-order statistic and thus its use wastes a lot of data. The chapter describes two modeling approaches based, respectively, on the generalized Pareto distribution and the joint distribution of the r-largest-order statistics that make use of more of the data.