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Introduction to Extreme Value Theory (IntroExtremValue) (AAM)
Provider: Faculty of Science

Activity no.: 5576-21-07-31There are 50 available seats 
Enrollment deadline: 22/11/2021
PlaceDepartment of Mathematical Sciences
Universitetsparken 5, 2100 København Ø
Date and time22.11.2021, at: 08:00 - 30.01.2022, at: 16:00
Regular seats50
ECTS credits7.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserThomas Valentin Mikosch    E-mail address: mikosch@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Semester/BlockBlock 2
Scheme groupA (Tues 8-12 + Thurs 8-17)
Exam formOral examination, 30 min. Without time for preparation
Exam formContinuous assessment
Exam detailsContinuous assessment, Two written assignments 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.
Grading scale7 point grading scale. For PhD students: Passed / Not Passed
Course workload
Course workload categoryHours
Project work60.00


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.

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.

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.

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

Teaching and learning methods
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.

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