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

Symmetry

Michael D. Perlman¹ 

[1] Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA

关键词: bivariate distribution;    multivariate distribution;    unit disk;    unit ball;    circular symmetry;    spherical symmetry;    circular copula;    spherical copula;    elliptical copula;   

DOI  :  10.3390/sym3030574

来源: mdpi

【 摘 要 】

Do there exist circular and spherical copulas in ℝ^d? That is, do there exist circularly symmetric distributions on the unit disk in ℝ² and spherically symmetric distributions on the unit ball in ℝ^d, 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 ℝ^d are also described, and determined explicitly for d = 2.

【 授权许可】

CC BY   
This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.