Metric embeddings of tail correlation matrices
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SeriesErasmus Econometric Institute Series
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Speaker(s)Anja Janssen (Otto-von-Guericke-University Magdeburg, Germany)
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FieldEconometrics
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LocationErasmus University Rotterdam, E building, room ET-14
Rotterdam -
Date and time
March 28, 2024
12:00 - 13:00
Abstract
The assessment of risks associated with multivariate random vectors relies heavily on understanding their extremal dependence, crucial in evaluating risk measures for financial or insurance portfolios. A widely used metric for assessing tail risk is the tail correlation matrix of tail correlation coefficients. Among the exploration of structural properties of the tail correlation matrix the so-called realization problem of deciding whether a given matrix is the tail correlation matrix of some underlying random vector has recently received some attention.
The entries of the tail correlation matrix are closely related to a useful distance measure on the space of Frechet-random variables, named spectral distance and first introduced in Davis & Resnick (1993). We analyze the properties of a related semimetric and show that it has the special property of being embeddable both in vector and function space, equipped with the respective sum norm. Notably, these embeddings bear a direct relationship with the realization of specific tail dependence structures via max-stable random vectors. Particularly, an embedding in vector space, employing so-called line metrics, provides a representation through a max-stable mixture of so-called Tawn-Molchanov models, s. also Fiebig, Strokorb & Schlather (2017).
Leveraging this framework, we revisit the realization problem, affirming a conjecture by Shyamalkumar & Tao (2020) regarding its NP-completeness.
This talk is based on joint work with Sebastian Neblung (University of Hamburg) and Stilian Stoev (University of Michigan).