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Symmetry 2011, 3(3), 574-599; doi:10.3390/sym3030574
Article

Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas

*  and
Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA
* Author to whom correspondence should be addressed.
Received: 30 December 2010 / Revised: 10 August 2011 / Accepted: 15 August 2011 / Published: 23 August 2011
(This article belongs to the Special Issue Symmetry in Probability and Inference)
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Abstract

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.
Keywords: bivariate distribution; multivariate distribution; unit disk; unit ball; circular symmetry; spherical symmetry; circular copula; spherical copula; elliptical copula bivariate distribution; multivariate distribution; unit disk; unit ball; circular symmetry; spherical symmetry; circular copula; spherical copula; elliptical copula
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Perlman, M.D.; Wellner, J.A. Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas. Symmetry 2011, 3, 574-599.

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