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Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas.
Michael D Perlman1, Jon A Wellner
1Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA.
Symmetry
|June 12, 2012
Summary
Circular and spherical copulas exist in dimensions 2 and 3, but not in higher dimensions (d ≥ 4). These unique copulas are explicitly determined, with extensions to elliptical and non-linear transformations.
Area of Science:
- Probability theory
- Multivariate statistics
- Copula theory
Background:
- Copulas are essential for modeling multivariate dependence.
- Circular and spherical symmetry are key properties in probability distributions.
- The existence of such copulas with uniform marginals is an open question for dimensions d ≥ 3.
Purpose of the Study:
- To investigate the existence of circular and spherical copulas in various dimensions.
- To determine if these copulas can have uniform one-dimensional marginal distributions.
- To explore transformations of these copulas.
Main Methods:
- Analysis of circularly symmetric distributions on the unit disk.
- Analysis of spherically symmetric distributions on the unit ball.
- Application of oblique coordinate transformations.
- Investigation of non-linear transformations of uniform distributions.
Main Results:
- Circular and spherical copulas with uniform marginals exist and are unique for dimensions d = 2 and 3.
- No such copulas exist for dimensions d ≥ 4.
- A one-parameter family of elliptical bivariate copulas was derived.
- Copulas from non-linear transformations of uniform distributions were described and determined for d = 2.
Conclusions:
- The existence of circular and spherical copulas is dimension-dependent.
- Explicit constructions are provided for dimensions 2 and 3.
- The study provides insights into the limitations and extensions of these copula types.
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