{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:37:23Z","timestamp":1760243843623,"version":"build-2065373602"},"reference-count":9,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2011,8,23]],"date-time":"2011-08-23T00:00:00Z","timestamp":1314057600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Do there exist circular and spherical copulas in \u211dd? That is, do there exist circularly symmetric distributions on the unit disk in \u211d2 and spherically symmetric distributions on the unit ball in \u211dd, d \u2265 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 \u2265 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in \u211d2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in \u211dd are also described, and determined explicitly for d = 2.<\/jats:p>","DOI":"10.3390\/sym3030574","type":"journal-article","created":{"date-parts":[[2011,8,24]],"date-time":"2011-08-24T07:43:32Z","timestamp":1314171812000},"page":"574-599","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas"],"prefix":"10.3390","volume":"3","author":[{"given":"Michael D.","family":"Perlman","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA"}]},{"given":"Jon A.","family":"Wellner","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Washington, Box 354322, Seattle, WA 98195-4322, USA"}]}],"member":"1968","published-online":{"date-parts":[[2011,8,23]]},"reference":[{"key":"ref_1","unstructured":"Nelsen, R.B. (2006). An Introduction to Copulas, Springer. [2nd ed.]."},{"key":"ref_2","unstructured":"Feller, W. (1971). An Introduction to Probability Theory and Its Applications, John Wiley & Sons Inc.. [2nd ed.]."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/S0167-7152(00)00191-7","article-title":"On some characterizations of spherical distributions","volume":"54","year":"2001","journal-title":"Statist. Probab. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"811","DOI":"10.2307\/1968466","article-title":"Metric spaces and completely monotone functions","volume":"39","author":"Schoenberg","year":"1938","journal-title":"Ann. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/0047-259X(81)90082-8","article-title":"On the theory of elliptically contoured distributions","volume":"11","author":"Cambanis","year":"1981","journal-title":"J. Multivar. Anal."},{"key":"ref_6","unstructured":"Bracewell, R.N. (1986). The Fourier Transform and Its Applications, McGraw-Hill Book Co.. [3rd ed.]."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1572","DOI":"10.1110\/ps.8701","article-title":"How common is the funnel-like energy landscape in protein-protein interactions?","volume":"10","author":"Tovchigrechko","year":"2001","journal-title":"Protein Sci."},{"key":"ref_8","unstructured":"Gooch, B., and Sloan, P.P.J. (May, January 30). Ambient aperture lighting. Proceedings of the 2007 Symposium on Interactive 3D Graphics, SI3D 2007, Seattle, WA, USA."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Perlman, M.D., and Wellner, J.A. (2010). Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas, University of Washington, Department of Statistics. Available at http:\/\/www.stat.washington.edu\/research\/reports\/2011\/tr578.pdf.","DOI":"10.3390\/sym3030574"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/3\/574\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:57:09Z","timestamp":1760219829000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/3\/3\/574"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,8,23]]},"references-count":9,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2011,9]]}},"alternative-id":["sym3030574"],"URL":"https:\/\/doi.org\/10.3390\/sym3030574","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2011,8,23]]}}}