Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929199297593344 |
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| author | Hallin, Marc Konen, Dimitri |
| author_facet | Hallin, Marc Konen, Dimitri |
| contents | Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L1 loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02499 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours Hallin, Marc Konen, Dimitri Statistics Theory Probability Quantiles are a fundamental concept in probability and theoretical statistics and a daily tool in their applications. While the univariate concept of quantiles is quite clear and well understood, its multivariate extension is more problematic. After half a century of continued efforts and many proposals, two concepts, essentially, are emerging: the so-called (relabeled) geometric quantiles, extending the characterization of univariate quantiles as minimizers of an L1 loss function involving the check functions, and the more recent center-outward quantiles based on measure transportation ideas. These two concepts yield distinct families of quantile regions and quantile contours. Our objective here is to present a comparison of their main theoretical properties and a numerical investigation of their differences. |
| title | Multivariate Quantiles: Geometric and Measure-Transportation-Based Contours |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2401.02499 |