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

Volume 2013/2014
Education
MSc Programme in Acturial Mathematics
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.
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.
P. Embrechts, C. Klueppelberg, T. Mikosch. Modelling Extremal Events for Finance and Insurance. Springer, Heidelberg, 1997.  9th Printing 2012
Basic knowledge of probability theory, statistics and stochastic precesses.
5 hours of lectures per week for 7 weeks.
  • Category
  • Hours
  • Exam
  • 66
  • Lectures
  • 35
  • Theory exercises
  • 105
  • Total
  • 206
Credit
7,5 ECTS
Type of assessment
Oral examination, 30 minutes
Written assignment
The oral examination counts for 70% of the grade. The remaining 30% correspond to
a Mid Term (15%) and a Final Term Test (15%). In these take home tests, the student will solve some theoretical problems and get estimation experience with simulated and real-life financial and insurance data.
Exam registration requirements
The student must have received more than 50% of the marks for the Mid Term and the Final Terms Tests.
Aid
Without aids
Marking scale
7-point grading scale
Censorship form
No external censorship
Several internal examiners
Re-exam
Oral examination (30 minutes) with internal censor without preparation time and notes.
The student is admitted to the re-examination if he/she has received more than 50% of the marks in both the Mid Term and Final Term Tests. If this has not been achieved at the time of the first examination the student may resubmit the two tests before the re-examination.
Criteria for exam assesment

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