Introduction to Extreme Value Theory (IntroExtremValue)(AAM)

In this course the student will learn about the basics of modern extreme value theory.
These include the classical asymptotic theory about the weak limits of standardized maxima and order statistics (Generalized Extreme Value Distribution) and of the excesses above high thresholds (Generalized Pareto Distribution) for sequences of iid random variables. An important part occupies the classification of distributions in different Maximum Domains of Attraction of the limiting extreme value distributions. Based on this theory, statistical tools and methods for detecting extremes and estimating their distributions are considered. These include estimators
of the tail index of a Pareto-like distribution,  the extreme value index of a distribution, the parameters of an extreme value distribution  and  the estimation of high/low quantiles of a distribution and tail probabilities,  possibly outside the range of the data. We discuss notions such as Value-at-Risk and Expected Shortfall which ate relevant for Quantitative Risk Management and their relation with extreme value theory. In the end of course, we discuss how the classical theory for independent variables can be extended to dependent observations. Such observations typically contain clusters of extreme values. We will learn about the extremal index which measures the size of a cluster and about the extremogram which measures lag-wise extremal dependence in a time series. The theory will be illustrated by various data sets from finance, insurance and telecommunications.