A Brief Introduction to Statistical Extreme Value Theory

Abstract

An important question to ask before applying EVT to a particular problem is when does it apply in a formal sense? Fundamentally, the answer is when the distribution to be modeled consists of extrema. As emphasized above in Chapter 1, extrema are the minima or maxima sampled from an overall distribution of data. To quote Coles [2001] “The distinguishing feature of an extreme value analysis is the objective to quantify the stochastic behavior of a process at unusually large—or small—levels.” Assume a sequence of i.i.d. samples (s₁; s₂;…. The maximum over an n-observation period is thus:

$$M_{n}=max(s_{1},s_{2},...).\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (2.1)$$

For large values of n, the approximate behavior of M_n follows from the limit arguments associated with n approaching infinity. From this observation, an entire family of models can be calibrated via the observed extrema values of M_n.