Modelling of multivariate extreme data

English: The aim of this thesis is to investigate the modelling of multivariate extreme values. Extreme value theory is becoming very popular as a statistical discipline. Not only is extreme value theory emerging in the statistical field but also in other disciplines such as engineering, financial markets and energy markets. Extreme value analysis focuses on the probability of unusual or extreme events. Extreme value theory is particularly useful in environmental applications such as the modelling of high rainfall, strong wind, etc. A lot of literature is available on the modelling of univariate extreme values. One topic of interest is the calculation of probabilities of related extreme events. To address this topic the modelling of multivariate extreme values is investigated. Various models are considered for modelling multivariate extreme values. The data set used in the thesis is the daily inflow of water, in cubic meters per second, into the Gariep Dam over a period of 29 years, from 1976 to 2006, excluding 1980 and 1982 due to large data losses. The models considered are the Multivariate Generalized Burr-Gamma distribution (Chapter 2), multivariate regression (Chapter 3), the Gumbel, Tawn and Logistic copulas (Chapter 5 and 6), and the Dirichlet mixture model (Chapter 5). The emphasis of Extreme Value Theory lies in the extreme values or events. Therefore, a threshold is chosen and only data above the threshold is modelled. Hence, specific research is done in this thesis on how to choose a threshold (Chapter 4). The preferred threshold method in this thesis is the method of the negative differential entropy of the Dirichlet process (Chapter 4). The thesis mainly considers a Bayesian approach for the estimation of parameter, although other approaches are also discussed. The last chapter, Chapter 7, gives a conclusion on the work that is covered in the thesis and recommendations for further research.