Monge-Kantorovich superquantiles and expected shortfalls with applications to multivariate risk measurements
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866911999723569152 |
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| author | Bercu, Bernard Bigot, Jeremie Thurin, Gauthier |
| author_facet | Bercu, Bernard Bigot, Jeremie Thurin, Gauthier |
| contents | We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_01584 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Monge-Kantorovich superquantiles and expected shortfalls with applications to multivariate risk measurements Bercu, Bernard Bigot, Jeremie Thurin, Gauthier Statistics Theory We propose center-outward superquantile and expected shortfall functions, with applications to multivariate risk measurements, extending the standard notion of value at risk and conditional value at risk from the real line to $\mathbb{R}^d$. Our new concepts are built upon the recent definition of Monge-Kantorovich quantiles based on the theory of optimal transport, and they provide a natural way to characterize multivariate tail probabilities and central areas of point clouds. They preserve the univariate interpretation of a typical observation that lies beyond or ahead a quantile, but in a meaningful multivariate way. We show that they characterize random vectors and their convergence in distribution, which underlines their importance. Our new concepts are illustrated on both simulated and real datasets. |
| title | Monge-Kantorovich superquantiles and expected shortfalls with applications to multivariate risk measurements |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2307.01584 |