Introduction to Extreme Value Theory (IntroExtremValue)(AAM)
Practical information  

Study year  2016/2017 
Time 
Block 1 
Programme level  Full Degree Master 
ECTS  7,5 ECTS 
Course responsible  
Phone: +45 35 320776
Office. 04.3.04  

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 Paretolike 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.
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, tyear 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.
Recommended prerequisites
"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.
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Education
MSc Programme in Acturial Mathematics
Studyboard
Study Board of Mathematics and Computer ScienceCourse type
Single subject courses (day)Duration
1 blockSchedulegroup
Teaching and learning methods
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 reallife financial and insurance data.
Capacity
No limitLanguage
EnglishWorkload
Category  Hours 
Lectures  35 
Theory exercises  105 
Exam  66 
English  206 
Exam
Type of assessment
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
7point grading scaleCriteria for exam assessment
The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.
Censorship form
No external censorshipReexam
Oral examination (30 minutes) with internal censor without
preparation time and without aids.
The student is admitted to the reexamination if he/she has
received more than 50% of the marks in both the Mid Term and Final
Term written assignments. If this has not been achieved at the time
of the first examination the student may resubmit the two written
assignments no later than two weeks before the beginning of the
reexamination week. The written assignments must be approved
before the reexam.
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