# 401-4912-15L Multivariate Extreme Value Theory and Max-Stable Processes

Semester | Spring Semester 2015 |

Lecturers | E. Koch |

Periodicity | non-recurring course |

Language of instruction | English |

Abstract | This course provides an introduction into the multi-dimensional and infinite-dimensional theories of extremes. This is done both from a probabilistic as well as a statistical point of view. |

Objective | For the modeling of many types of risks (as environmental ones), the univariate theory of extremes is insufficient since it does not allow to account for the dependence between different variables or different locations. In this course, students learn some theoretical concepts and practical tools about the dependence of extremes of different variables or of the same variable but at different sites (multivariate setting). The case of an infinite number of locations (continuous setting) i.e. the spatial extremes context, is also considered. The course provides the main probabilistic results as well as some modelling and data-analysis methods. Concrete examples are developed. |

Content | Introduction and preliminaries: - Introduction to extreme events; - Motivation about the multivariate and the spatial setting; - Preliminaries about point processes; - Univariate extreme value theory: Main probabilistic results i.e. the Fisher-Tippett Theorem and the Pickhands-Balkema-de Haan Theorem; Multivariate setting: - Max-stable distributions; - Different representations of the multivariate extreme value distribution; - Link with extreme value copulas; - Exponent measure and spectral measure; - Parametric families and inference; - Asymptotic dependence and independence; - Multivariate maximum domain of attraction; Infinite-variate setting i.e. spatial extremes: - Definition of max-stable processes; - Spectral representation of max-stable processes and parametric models; - Inference methods for max-stable models; - Some applications to environmental risk analysis. |

Literature | Books: 1. S.G. Coles (2001) An Introduction to Statistical Modeling of Extreme Values. Springer; 2. J. Beirlant, Y. Goegebeur, J. Segers and J.L. Teugels (2004) Statistics of Extremes: Theory and Applications, Wiley. 3. P. Embrechts, C. Klueppelberg and T. Mikosch (1997) Modelling Extremal Events for Insurance and Finance. Springer. 4. L. de Haan and A. Ferreira (2006) Extreme Value Theory. An Introduction. Springer. 5. S.I. Resnick (1987) Extreme Values, Regular Variation, and Point Processes. Springer. Papers: 1. L. de Haan (1984) A spectral representation for max-stable processes. The Annals of Probability, 12(4):1194-1204. 2. S.A. Padoan, M. Ribatet and S.A. Sisson (2010). Likelihood-based inference for max-stable processes. Journal of the American Statistical Association, 105(489):263-277. 3. J. Pickands (1981). Multivariate extreme value distributions. In Proceedings 43rd Session International Statistical Institute, 2:859-878. 4. M. Schlather (2002) Models for stationary max-stable random fields. Extremes, 5(1):33-44. 5. R.L. Smith (1990). Max-stable processes and spatial extremes. Unpublished manuscript, University of North Carolina. |