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Title: Determining the dependence structure of multivariate extremes Authors:  Emma Simpson - Lancaster University (United Kingdom) [presenting]
Jenny Wadsworth - Lancaster University (United Kingdom)
Jonathan Tawn - Lancaster University (United Kingdom)
Abstract: When modelling multivariate extremes, it is important to consider the asymptotic dependence properties of the variables. This can have a complicated structure, with only certain subsets of variables taking their largest values simultaneously. We will discuss a method that, given a set of data, aims to establish this asymptotic dependence structure. It is a common method in multivariate extreme value analysis to consider variables in terms of their radial and angular components $(R, \mathbf{W})$. In this case, the angular component $\mathbf{W}$ takes values in the unit simplex and, conditioning on $R$ being extreme, the position of mass on this simplex can reveal the asymptotic dependence structure of the variables of interest. In the bivariate case, this corresponds to deciding whether or not there is mass on the interior or the edges of the simplex, but in the $d$-dimensional case there are $2^{d}-1$ sub-simplices that could contain mass. In reality, data will not lie exactly on the sub-simplices of the angular unit simplex, so assessing the asymptotic dependence structure is not a straightforward task. By partitioning the angular simplex, we aim to find the conditional probability that a point lies in a certain section given that it is extreme in terms of its radial component. This allows us to determine the asymptotic dependence structure of the variables, as well as the proportion of mass on each sub-simplex asymptotically.