[Matematisk statistik] [Matematikcentrum] [Lunds tekniska högskola] [Lunds universitet]
[Övriga kurser]

# FMSN15/MASM23: Statistical Modeling of Multivariate Extremes

## General information:

• Credits: 7.5hp/7.5 ECTS credits.

• Study period: Period 2.

• Language: The course will be given in English.

• Requirements: FMS155/MASM15: Statistical Modeling of Extreme Values

• Course Description:
Extreme value theory has been subject of much practical and theoretical work in the last few years. Recent developments have introduced very flexible and theoretically well motivated semi-parametric models for extreme values which now are at the stage where they can be used to address important technological problems on handling risks in areas such as wind engineering, hydrology, flood monitoring and prediction, climatic changes, structural reliability, corrosion modelling, and large insurance claims or large fluctuations in financial data (volatility).

The course has three main objectives which are summarized below.

• We will give an introduction to copulas and dependence modeling in general. This will include Sklar's theorem, the Frechet-Hoeffding bounds for joint distributions, simulation of copulas, Kendall's tau, Spearman's rho and other measures of association. We will also discuss elliptical distributions, Archimedean Copulas, upper and lower tail dependence and will give examples of tail dependence for spherical and elliptical copulas and some parametric families of copulas.

• We will cover the probability theory of multivariate extreme value theory in the independent case. Our approach will be mainly from applied point of view so statistical modeling of extreme events will be main emphasis of the course. This includes weak convergence for normalized extreme values of stochastic vectors, different characterizations of multivariate extreme value distributions, "peaks over threshold"-model in the multivariate case, different definitions of multivariate generalized Pareto distributions, statistical inference for multivariate extreme values, parametric and semi-parametric methods for multivariate extreme values, use of copula in modeling extreme values, prediction of extreme values, examples of applications of the theory, e.g., estimation of operational risk, climate changes and wind insurances.

• As a modeling language we will be using R so, upon completing the course, participants will be able to use and write programs in R for their own applications. To this end we will discuss a few examples of applications of the extreme value theory in R in details. These example are mainly taken from a few papers which have recently been published and we will go through the details of calculations which are behind the results presented in the papers. We will also cover the fundamentals of R which are necessary for statistical modeling in general. This will include: overview of R, data objects in R, data import and export, manipulation and restructuring, graphics, functions and operators, writing functions and optimization routines in R. In addition some specialized libraries for using copulas and analyzing multivariate extreme value data will be discussed.

• Course syllabus

### Literature

During the course we will use material presented in various papers. The following books will be the main references:

1. An Introduction to Copulas. Roger B. Nelsen. This book is available as e-book here.
2. Statistics of Extremes: Theory and Applications. Jan Beirlant, Yuri Goegebeur, Johan Segers, Jozef Teugels, with contributions from Daniel De Waal, Chris Ferro. This book is available as a reference book at Mathematics library. It is also available to the students and staff of Lund University as ebook at Wiley Series in Probability and Statistics.