Classifying and exploiting structure in multivariate extremes

Simpson, Emma and Wadsworth, Jennifer and Tawn, Jonathan (2019) Classifying and exploiting structure in multivariate extremes. PhD thesis, Lancaster University.

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Abstract

The aim of this thesis is to present novel contributions in multivariate extreme value analysis, with focus on extremal dependence properties. A set of random variables is often categorized as being asymptotically dependent or asymptotically independent, based on whether the largest values occur concurrently or separately across all members of the set. However, there may be a more complicated structure to the extremal dependence that is not fully described by this classification, with different subsets of the variables potentially taking their largest values simultaneously while the others are of smaller order. Knowledge of this detailed structure is essential, and aids efficient statistical inference for rare event modelling. We propose a new set of indices, based on a regular variation assumption, that describe the extremal dependence structure, and present and compare a variety of inferential approaches that can be used to determine the structure in practice. The first approach involves truncation of the variables, while in the second we study the joint tail behaviour when subsets of variables decay at different rates. We also consider variables in terms of their radial-angular components, presenting one method based on a partition of the angular simplex, alongside two soft-thresholding approaches that incorporate the use of weights. The resulting estimated extremal dependence structures can be used for dimension reduction, and aid the choice or construction of appropriate extreme value models. We also present an extensive analysis of the multivariate extremal dependence properties of vine copulas. These models are constructed from a series of bivariate copulas according to an underlying graphical structure, making them highly flexible and useful in moderate or even high dimensions. We focus our study on the coefficient of tail dependence, which we calculate for a variety of vine copula classes by applying and extending an existing geometric approach involving gauge functions. We offer new insights by presenting results for trivariate vine copulas constructed from bivariate extreme value and inverted extreme value copulas. We also present new theory for a class of higher dimensional vine copulas. An approach for predicting precipitation extremes is presented, resulting from participation in a challenge at the 2017 EVA conference. We propose using a Bayesian hierarchical model with inference via Markov chain Monte Carlo methods and spatial interpolation.