# 401-3919-60L An Introduction to the Modelling of Extremes

Semester | Spring Semester 2016 |

Lecturers | P. Embrechts |

Periodicity | yearly course |

Language of instruction | English |

Abstract | This course yields an introduction into the one-dimensional theory of extremes, and this both from a probabilistic as well as statistical point of view. This course can be seen as a first course on extremes, a sequel concentrating more on multivariate extremes. |

Objective | In this course, students learn to distinguish between so-called normal models, i.e. models based on the normal or Gaussian distribution, and so-called heavy-tailed or power-tail models. They learn to do some standard modelling and data analysis for one-dimensional data. The probabilistic key theorems are the Fisher-Tippett Theorem and the Balkema-de Haan-Pickands Theorem. These lead to the statistical techniques for the analysis of extremes or rare events known as the Block Method, and Peaks Over Threshold method, respectively. |

Content | - Introduction to rare or extreme events - Regular Variation - The Convergence to Types Theorem - The Fisher-Tippett Theorem - The Method of Block Maxima - The Maximal Domain of Attraction - The Fre'chet, Gumbel and Weibull distributions - The POT method - The Point Process Method: a first introduction - The Pickands-Balkema-de Haan Theorem and its applications - Some extensions and outlook |

Lecture notes | There will be no script available. |

Literature | At a more elementary level: [1] S.G. Coles (2001) An Introduction to Statistical Modeling of Extreme Values. Springer. [2] R.-D. Reiss and M. Thomas (1997) Statistical Analyis of Extreme Values. Birkhaeuser. At an intermediate level: [3] J. Beirlant, Y. Goegebeur, J. Segers and J.L. Teugels (2004) Statistics of Extremes: Theory and Applications, Wiley. [4] P. Embrechts, C. Klueppelberg and T. Mikosch (1997) Modelling Extremal Events for Insurance and Finance. Springer. [5] S. I. Resnick (2007) Heavy-Tail Phenomena. Probabilistic and Statistical Modeling. Springer. At a more advanced level: [6] L. de Haan and A. Ferreira (2006) Extreme Value Theory. An Introduction. Springer. [7] S. I. Resnick (1987) Extreme Values, Regular Variation, and Point Processes. Springer. |