Extreme value analysis based in part on the article “Extreme value analysis” by Saralees Nadarajah, which appeared in the encyclopaedia of environmetrics.

Extreme value theory concerns the behavior of the extremes of a process or processes. The fundamentals of this probability theory have been known since about the beginning of the twentieth century, but the relevant statistical methods for modeling extreme values emerged in the literature only in the past three decades. In fact, since 1980 the literature has seen a flood of applications of statistical extreme values, covering a wide range of areas. These application areas include environmental sciences, including climate, engineering and hydrology, performance assessment as in sports or policing, astronomy, finance, chemometrics, mortality studies, and outlier detection. Further references to specific applications are noted throughout this article. The aim of this article is to review some fundamentals of extreme value theory and the relevant statistical methods. The emphasis will be on the latter and the applications it has attracted in the literature so far. The article is in two parts. The first part considers univariate extremes and the remainder is for multivariate extremes. Each part begins with a discussion of fundamental theoretical results. This is then followed by a discussion of relevant statistical models, inference, and simulation.