Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas
Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, w...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2011-08-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/3/3/574/ |
| _version_ | 1811301777294753792 |
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| author | Michael D. Perlman Jon A. Wellner |
| author_facet | Michael D. Perlman Jon A. Wellner |
| author_sort | Michael D. Perlman |
| collection | DOAJ |
| description | Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝd are also described, and determined explicitly for d = 2. |
| first_indexed | 2024-04-13T07:14:56Z |
| format | Article |
| id | doaj.art-d6a1a7ceada147eb813a84f59fafb73f |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-13T07:14:56Z |
| publishDate | 2011-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-d6a1a7ceada147eb813a84f59fafb73f2022-12-22T02:56:46ZengMDPI AGSymmetry2073-89942011-08-013357459910.3390/sym3030574Squaring the Circle and Cubing the Sphere: Circular and Spherical CopulasMichael D. PerlmanJon A. WellnerDo there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝd are also described, and determined explicitly for d = 2.http://www.mdpi.com/2073-8994/3/3/574/bivariate distributionmultivariate distributionunit diskunit ballcircular symmetryspherical symmetrycircular copulaspherical copulaelliptical copula |
| spellingShingle | Michael D. Perlman Jon A. Wellner Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas Symmetry bivariate distribution multivariate distribution unit disk unit ball circular symmetry spherical symmetry circular copula spherical copula elliptical copula |
| title | Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas |
| title_full | Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas |
| title_fullStr | Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas |
| title_full_unstemmed | Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas |
| title_short | Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas |
| title_sort | squaring the circle and cubing the sphere circular and spherical copulas |
| topic | bivariate distribution multivariate distribution unit disk unit ball circular symmetry spherical symmetry circular copula spherical copula elliptical copula |
| url | http://www.mdpi.com/2073-8994/3/3/574/ |
| work_keys_str_mv | AT michaeldperlman squaringthecircleandcubingthespherecircularandsphericalcopulas AT jonawellner squaringthecircleandcubingthespherecircularandsphericalcopulas |