19 - Extreme Value Theory

Summary

This chapter gives an introduction to extreme value theory. Unlike most statistical analyses, which are concerned with the typical properties of a random variable, extreme value theory is concerned with rare events that occur in the tail of the distribution. The cornerstone of extreme value theory is the Extremal Types Theorem. This theorem states that the maximum of N independent and identically distributed random variables can converge, after suitable normalization, only to a single distribution in the limit of large N. This limiting distribution is called the Generalized Extreme Value (GEV) distribution. This theorem is analogous to the central limit theorem, except that the focus is on the maximum rather than the sum of random variables. The GEV provides the basis for estimating the probability of extremes that are more extreme than those that occurred in a sample. The GEV is characterized by three parameters, called the location, scale, and shape. A procedure called the maximum likelihood method can be used to estimate these parameters, quantify their uncertainty, and account for dependencies on time or external environmental conditions.