Course Syllabus for

# Statistical Modelling of Multivariate Extreme Values Statistisk modellering av multivariata extremvĂ¤rden

## FMSN15F, 7.5 credits

Valid from: Autumn 2020
Decided by: Professor Thomas Johansson
Date of establishment: 2020-05-19

## General Information

Division: Mathematical Statistics
Course type: Course given jointly for second and third cycle
The course is also given at second-cycle level with course code: FMSN15
Teaching language: English

## Aim

Multivariate extreme values occure in, e.g., economy, safety and reliability, insurance mathematics, hydrology, meteorology. environmental sciences, och ocenanography. They often show complicated dependencies between several variables, e.g. between wind speed, wind direction, wave height and ocean currents. This calls for special methods that can be used, e.g., for analysis of trends, calculation of flooding risks, and modelleling storm damage, corrosion speed, or financial risks. Climat and environmental changes, as well as an increasingly complicated financial market, pose new demands on deapend knowledge in these fields. This course is a countinuation of FMSN55 Statistical Modelling of Extreme Values, and teaches methods for analysis of multivariate and spatial extreme values.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• describe how to define extreme values for multivariate samples,
• describe different characterisations of multivariate extreme value distributions and the relationship between them,
• explain how to generalize the "peaks over threshold"-model to higher dimensions and which asymptotic distributions arise,
• explain which statistical methods can be used for the analysis of extreme values.

Competences and Skills

For a passing grade the doctoral student must

• handle multivariate data for analysis of extreme values,
• fit extreme value distribution using different methods,
• validate the valitidy of the extreme value model and make suitable modifications of the model,
• use the resulting model for prediction,
• use a statistical computer program for analysis of data,
• present the analysis and conclusions of a practical problem in a written report.

Judgement and Approach

For a passing grade the doctoral student must

• always check the prerequisites befor stating an extreme value model,
• evaluate the plausibility of a performed study,
• reflect over the limitations of the chosen model and estimation method, as well as alternative solutions.

## Course Contents

Weak convergence for normalized extreme values of stochastic vectors, different characterisations 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 modelling extreme values, point process characterisation of extreme values, prediction of extreme values, examples of applications of the theory, e.g., estimation of operational risk, climate changes and wind insurances.

## Course Literature

• Beirlant, J., Goegebeur, Y., Segers, J. & Teugels, J.: Statistics of Extremes: Theory and Applications. Wiley, 2004.
• Nelson, Roger B.: An Introduction to Copulas. Springer, 2006.

## Instruction Details

Types of instruction: Lectures, laboratory exercises, exercises

## Examination Details

Examination formats: Written exam, written assignments
Grading scale: Failed, pass
Examiner: Senior lecturer Nader Tajvidi

## Admission Details

Admission requirements: FMSN55 Statistical Modelling of Extreme Values

## Course Occasion Information

Start date: 2021-03-22
Course pace: Half time

## Contact and Other Information

Course coordinator: Nader Tajvidi <nader.tajvidi@matstat.lu.se>
Web page: www.maths.lth.se/matstat/kurser/fmsn15/