### MATH-447 Risk, rare events and extremes #### SUMMARY

The statistical modelling of rare events, such as stock market crashes, storms and catastrophic structural failures, is increasingly seen to be important. For example, the most destructive effects of climatic change are likely to be due to changes in extreme events, and events in the financial markets are also often modelled using methods from extreme-value statistics. This course will describe the special models and methods that are relevant to the analysis of such data, including the mathematical bases, statistical tools, and practical applications.

#### CONTENT

• Mathematical bases: behaviour of maxima and threshold exceedances in large samples, both for independent and dependent data. Poisson process modelling.
• Statistical methods: modelling using the GEV and GP distributions, for independent and dependent data. Likelihood and Bayesian inference. Non-stationarity. Extremal coefficients. Multivariate extreme-value distributions. Max-stable processes.
• Applications: Environmental, financial, and engineering applications. Use of R for extremal modelling.

#### LEARNING PREREQUISITES

##### Important concepts to start the course

Probability and statistics at the level of second-year bachelor (mathematics), plus further knowledge of statistics and stochastic processes.

#### LEARNING OUTCOMES

By the end of the course, the student must be able to:
• Recognize situations where statistical analysis of extrema is appropriate
• Manipulate mathematical objects related to the study of extrema
• Analyze empirical data on extremes using appropriate statistical methods
• Construct appropriate statistical models for extremal data
• Interpret such models in terms of underlying phenomena
• Infer properties of real systems in terms of probability models for extremes

#### TEACHING METHODS

Lectures, theoretical and computational exercises in class and at home.

#### ASSESSMENT METHODS

Mini-project, final exam.

#### RESOURCES

##### Bibliography

Coles, S. G. (2001) An Introduction to the Statistical Modelling of Extreme Values. Springer.

Beirlant, J, Goegebeur. Y., Teugels. J. and Segers. J. (2004) Statistics of Extremes: Theory and Applications. Wiley.