A Geometric Interpretation of Multivariate Extreme Value Analysis

Campbell, Ryan and Wadsworth, Jennifer (2025) A Geometric Interpretation of Multivariate Extreme Value Analysis. PhD thesis, Lancaster University.

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Abstract

In multivariate extreme value analysis, interest lies in characterising the tail behaviour of random vectors. The lack of natural ordering in the multivariate setting, and the many possible combinations of tail behaviour for all the subgroups of components of random vectors make this a difficult task. In this work, the geometry of scaled copies of random vectors is used to characterise the tail of the underlying probability distribution. While this geometry has been shown to provide useful information about the tail behaviour of random vectors, we introduce methodology to estimate it from data in a parametric, Bayesian semiparametric, and piecewise-linear semiparametric manner. The geometry is used to model both the radial and angular components of the pseudo-polar decomposition, a key feature of the geometric framework. Links are made to the classical approach of multivariate extremes by investigating the geometry of generalised Pareto random vectors, an important model used in a variety of practical applications. Both the geometric and the classical approach have their benefits and drawbacks. These will be discussed along with a commentary on future work to be done in the multivariate geometric framework.