Multivariate extreme value inference based on tail expansions of copulas with applications to systemic risk analysis

Extreme events are inherently rare, and this poses significant challenges for statistical modeling and inference as they require extrapolating beyond observed data. This thesis develops theoretically grounded probabilistic frameworks and statistical methods to facilitate such extrapolation in multivariate extreme inference. A central contribution of this work is the introduction of tail expansions of copulas, which provide a systematic approach to characterize the joint tail behavior of multiple dependent random variables. This characterization naturally integrates and extends classical extreme value theory, offering a more flexible and interpretable representation of extremal dependence. Tail expansions of copulas provide a new theoretical basis for extrapolating beyond observed data. Building on this framework, this work develops a suite of statistical tools, including estimation methods, dependence measures, diagnostic techniques, and modeling strategies, to enhance both the flexibility and reliability of extrapolation. These tools hold significant practical value in analyzing and modeling extreme dependence in real-world data. A primary motivation for these theoretical and methodological developments is to address empirical challenges in quantitative risk management, with a particular focus on assessing and mitigating systemic risk. This work examines the Conditional Value-at-Risk (CoVaR) through the lens of tail expansions of copulas, yielding both theoretical insights and practical estimation approaches. By integrating tail expansion with CoVaR, our approach improves the empirical analyses of systemic risk, enabling more precise quantification and a deeper understanding of its dynamics.