Extreme Value Theory

The main topics covered in this chapter are:what extreme value theory is and how it differs from classical statistics;the two pillars of extreme value theory: Fisher–Tippett–Gnedenko theorem and Pickands–Balkema–de Haan theorem;the three classes that the limit distribution of maxima will fall into: the Fréchet, Weibull, or Gumbel distribution;the generalized Pareto distribution;the maximum domain of attraction of an extreme value distribution and the concept of tail equivalence;the theory of maxima for stationary processes;extreme value theory for multivariate distributions;the role of copula in multivariate extreme value theory;the three types of copulas;three estimation methods for distributions: maximum likelihood estimation method, method of moments, and special estimators;the Hill estimator and the Pickands estimator for estimating the shape parameter of a distribution;use and limitations of the quantile plot (QQ-plot) for verifying statistical hypotheses by examining the degree of deviations of the linearity plot of a hypothesized distribution;three different approaches to compute widely-known risk measures (VaR and AVaR).