Multivariate extremes: theory and applications beyond the classical paradigm

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Multivariate extremes lie at the heart of many fields of application including finance and insurance, civil and reliability engineering, hydrology and environmental sciences. In a risk analysis context, it is not the expected or typical outcomes but the extreme and rare ones that are of greatest concern. The traditional approach to analyzing and modelling of multivariate extremes is based on the asymptotic theory for coordinate-wise maxima, referred to as classical extreme value theory (EVT). Although still relevant for a wide range of problems, this approach suffers from several shortcomings and limitations. This calls for alternative methodological formulations and approaches that go beyond the paradigm of the classical theory. Exploring these alternatives constitutes the main theme of the proposed research program. The first part of the proposal deals with the probabilistic development of multivariate EVT using a geometric approach, initiated in our recent work. In particular, it involves the analysis of multivariate extremes and extreme dependencies in the model and data via the shape of the level sets of the underlying probability density function and the global asymptotic shape of the associated sample clouds. The second part concerns the issues surrounding statistics of multivariate extremes. The main challenge in this context is obvious - it is the very nature of extreme and hence rare events, which leaves either too little or no data to conduct statistical inference. In this situation, asymptotic theory can be used to provide a basis for data extrapolation. The aim is to develop statistical methodology for the analysis of multivariate extremes based on the probabilistic results of the first part of the proposal. The area of multivariate extremes and in particular its statistical estimation side is still in the process of active development, which started just a couple of decades ago. The results and methods to be investigated as part of the proposed research program are seen as a contribution towards maturity of this important and challenging area. The true significance of the proposed work will come in the form of the bridge between mathematical discoveries and their applications to the complex real life problems.