# Extreme value theory

## Time and place

**Lecture**

Thursday, 8:15 - 9:55, Heho18, E20 (Start: April 20th)

**Exercise classes**

Thursday, 16:15 - 17:55 (each 14 days), Heho 18, E60

## Type

2 hours lecture, 1 hour excercise classes

4 credit points

## Prerequisites

Stochastik 1

## Content

We will treat three topics:

### The limit distribution of the maximum

Consider independent and identical distributed random variables. Then, according to the Central Limit Theorem, the arithmetic mean converges (after a suitable linear transformation) to the normal distribution (provided that the variances of the random variables are positive and finite). But what happens if we replace the arithmetic mean by the maximum? It turns out that the limit distribution now belongs to one out of three families (which we will introduce in the lecture), namely to the Gumbel distributions to the Weibull distributions or to the Fréchet distributions.

### Statistical applications

Using the limit theorems from the first part, we can construct estimators that are e.g. helpful for the following problem:

We want to build a dike which is flooded only once in 200 years, but we have only water level data from the last 70 years. What should be the height of the dike?

Questions like this are also interesting for insurance companies.

### Outlook

We will also treat the following to questions:

- What can we say about the record times (in the example above a record is a time when a flood occures which is higher than any flood until that time)?
- What can we say about the limit distribution of other quantiles of the sample, e.g. the median or the third largest element?

## Lecture notes

Lecture notes (in german) will be put online step by step.

## Excercise sheets

Submitting the exercise sheets is necessary to achieve the prerequisite. You have to register for the excercise classes in SLC.

## Prerequisite and exam

Prerequisite are 50% of all exam credits.

There will be a written exam at the end of this term and the possibility to have an oral exam at the beginning of the next term.

### Students who have achieved the prerequisite earlier

Students who can proof using their transcripts of records that they have achieved the prerequisite for a **master** lecture "extreme value theory" earlier do not need to achieve it again. Watch out for further announcements. You will have to show your transcript of records in the end of June / in the beginning of July.

This does not apply for students who have achieved the prerequisite in 2015 since the lecture "extreme value theory" was a bachelor lecture in that year.

## Literature

Most recommended is

- P. Embrechts, C. Klüppelberg, T. Mikosch:
*Modelling Extremal Events for Insurance and Finance*. Springer, 2013.

There are many books that start similar as the lecture, but than turn to other topics (within the field of extreme value theory), e.g.:

- S. Resnick:
*Extreme Values, Regular Variation and Point Processes*. Springer, 2007. - L. de Haan, A. Ferreira:
*Extreme Value Theory: An Introduction*. Springer, 2006.

The following book has its focus on the statistical aspects of extreme value theory:

- J. Beirlant, Y. Goegebeur, J. Segers, J. Teugels:
*Statistics of Extremes: Theory and Applications*. Wiley, 2005.

# Contact

**Lecturer & Teaching assistant**

Helmholtzstraße 18, Raum 1.46

0731 50 23526

# News

**It has been decided that the course should be taught in german.**

**From now on only the german version of this page will be updated.**