# Introduction to Multivariate Extreme Value Theory (AAM)

### Course content

In this course we will get to know the mathematical tools for the handling of extremal events of multivariate random variables. This means that we will study random vectors (often of dimension 2) which typically show dependency among their entries. Think for example of the stock prices of two different assets or of the wave height and the still water level of the Baltic Sea at a certain point.

The severeness of an extremal event then often depends on the specific combination of outcomes of vector entries and we therefore seek for ways to describe a limiting distribution for the random vector given that at least one component is large. The mathematical tool for this analysis is multivariate regular variation and we will explore this concept in depth.

We will get to know different ways to describe the resulting limiting distribution like the stable tail dependence function, Pickands' dependence function and the exponent measure. Furthermore, we will describe different ways for the estimation of these quantities and how it can be used to approximate the probabilities for a specific extremal event. We will also look at practical examples and how the estimation procedures can be implemented with the help of statistical software (namely: R) and suitable packages.

Towards the end of the course, we will focus particularly on the case of asymptotically independent data, i.e. where typically only one of the components of the random vector is extreme. We will get to know refined methods of analysis and estimation (like the concept of regular variation on cones and the coefficient of tail dependence) which allow for non-trivial results for the extremal behavior of those vectors as well.

Education

MSc Programme in Actuarial Mathematics

Learning outcome

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

Knowledge:

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

• Theory of multivariate regular variation, max-stable distributions and multivariate extreme value distributions
• Methods for characterization of extremal dependence of two or more random variables
• Standard statistical methods and techniques for handling multivariate extreme values, like estimators for the stable tail dependence function and the spectral measure (both parametric and semiparametric)
• Methods for the analysis of asymptotically independent observations, in particular regular variation on cones and the coefficient of tail dependence
• Fields of applications for multivariate extreme value theory and statistics, i.e. analysis of financial assets.

Skills:

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

Competences:

The student will be competent in modeling extremes of multivariate random 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 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.

Knowledge from the course "Introduction to Extreme Value Theory" (Block 1) can be helpful but is not necessary. The necessary preconditions will be repeated during the course.

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 take home written assignment (15%).
Aid

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

Marking 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 Preparation
• 66
• English
• 206