# Introduction to Extreme Value Theory (IntroExtremValue)(AAM)

### Course content

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 limitng 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 have clusters of extreme values. We will learn about the extremal index which measures the size of clusters. The theory will be illustrated by various data sets
from finance, insurance and telecommunications.

Education

MSc Programme in Acturial Mathematics

Learning outcome

In this course, the student will learn about the basics of modern extreme value theory.

Knowledge:

In particular, he/she will know about the following topics:

Classical limit theory for sequences of iid observations and their excesses above high thresholds.

Exploratory statistical tools for detecting and classifying extremes.

Standard statistical methods and techniques for handling extreme values, including estimation for extreme value distributions and in their domains of attraction, the Peaks over Threshold (POT)  method for excesses above high thresholds.

Standard notions from Quantitative Risk Management  such as Value at Risk, Expected Shortfall, return period, t-year event, and their relation with extreme value theory.

The notion of cluster of extremes for dependent data and how to measure the size of clusters.

Skills:

At the end of the course,  the student will be able to read books, articles and journals
which are devoted to topics of modern extreme value theory and extreme value statistics.

Competences:

The student will be competent in modeling extremes of independent and weakly dependent observations and be able to apply software packages specialized for analyzing extreme values.

5 hours of lectures per week for 7 weeks.
In addition, two take home written assignments in which the student will solve some theoretical problems and get estimation experience with simulated and real-life financial and insurance data.

"Advanced Probability Theory 1 (VidSand1)" or a similar course is recommended for the necessary knowledge of probability theory and stochastic processes.
"Statistik 1 (Stat1)" or a similar course is recommended for the necessary knowledge of statistics.

ECTS
7,5 ECTS
Type of assessment
Oral examination, 30 minutes without preparation time
Written assignment
The oral examination counts for 70% of the grade.
The remaining 30% correspond to a Mid Term (15%) and a Final Term written assignment (15%).
Aid

The oral final exam is without aids. All aids are allowed for the two written assignments.

Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners
##### Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

• Category
• Hours
• Lectures
• 35
• Theory exercises
• 105
• Exam
• 66
• English
• 206