Stochastic ordering and sparse approximation of multivariate extremal dependence

Corradini, Michela 2024. Stochastic ordering and sparse approximation of multivariate extremal dependence. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

While traditional statistical methodologies focus on the average behaviour of a system and quantifying deviations from it, practical concern often lies on the extremal behaviour of a system. Understanding the statistics of extremes is crucial for answering questions arising from practical applications, such as the severity of floods or the extent of financial losses. Most real-world phenomena involve multiple variables, which may exhibit interaction at extreme levels; this motivates the study of multivariate extremes, the study of distributional tails of multivariate random vectors when two or more variables in the vector can be large together. This thesis addresses two challenges in multivariate extreme value theory. First, new (and highly non-trivial) stochastic orderings among multivariate extreme value distributions are revealed. More precisely, we consider the multi-variate stochastic orders of upper orthants, lower orthant and positive quadrant dependence (PQD) among simple max-stable distributions and their exponent measures. The main result shows that each of these orders holds for the max-stable distribution if and only if it holds for the corresponding exponent measure. Popular parametric models such as the Dirichlet and H¨usler-Reiß families are shown to be ordered according to the aforementioned multivariate stochastic orderings. Second, this thesis proposes a new method for estimating a sparse but accurate representation of the spectral measure, which contains the information about the dependence structure of multivariate extremes. In order to obtain such sparse approximations, we introduce techniques from the kernel mean embedding of measures to the context of spectral measure estimation in multivariate extremes. A broad range of numerical experiments shows that this is a promising approach.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Schools > Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EPSRC
Date of First Compliant Deposit:	25 April 2025
Last Modified:	25 Apr 2025 12:34
URI:	https://orca.cardiff.ac.uk/id/eprint/177916

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