# NMAK13005U Introduction to Extreme Value Theory (IntroExtremValue)(AAM)

Volume 2023/2024
Education

MSc Programme in Actuarial Mathematics
MSc Programme in Mathematics-Economics

MSc Programme in Statistics

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 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.

Learning Outcome

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

Knowledge:

In particular, the student 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.

Example of course literature:

C. Klueppelberg, P. Embrechts, T. Mikosch:

Modelling Extremal Events for Insurance and Finance.

Springer, 1997

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

Academic qualifications equivalent to a BSc degree is recommended.
5 hours of lectures per week for 7 weeks.
In addition, two take home written assignments (mid term and final term tests) in which the student will solve some theoretical problems and get estimation experience with simulated and real-life financial and insurance data.
• Category
• Hours
• Lectures
• 35
• Preparation
• 56
• Project work
• 60
• Exam Preparation
• 55
• Total
• 206
Oral

Oral feedback will be given on students’ presentations in class.

Feedback by final exam (in addition to the grade): in connection with oral exam and the two tests.

Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes without preparation time
Continuous assessment, Two written assignments
Type of assessment details
Two written assigments during the course: Mid Term and Final Term.
Oral exam (30 minutes) without preparation time.
The total grade will be determined by the overall results for both the assignments and the oral presentation. All parts of the exam must be passed in order to pass the course.
Aid
Only certain aids allowed

The oral final exam is without aids.

All aids are allowed for the two written assignments.

Marking scale
Censorship form
No external censorship
Several internal examiners for oral exam. One internal examiner for the written assignments.
Re-exam

Oral examination (30 minutes) with internal censors without preparation time and without aids.

If the two assignments were not passed in the ordinary examination, the student must re-submit the two written assignments no later than three weeks before the beginning of the re-examination week.

##### Criteria for exam assesment

In order to obtain the grade 12 the student should convincingly and accurately demonstrate the knowledge, skills and competences described under Learning Outcome.