{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T22:14:26Z","timestamp":1772748866463,"version":"3.50.1"},"reference-count":15,"publisher":"Oxford University Press (OUP)","issue":"5","license":[{"start":{"date-parts":[[2006,10,24]],"date-time":"2006-10-24T00:00:00Z","timestamp":1161648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2006,11,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>The complex Bingham distribution was introduced by Kent as a tractable model for landmark-based shape analysis. It forms an exponential family with a sufficient statistic which is quadratic in the data. However, the distribution has too much symmetry to be widely useful. In particular, under high concentration it behaves asymptotically as a normal distribution, but where the covariance matrix is constrained to have complex symmetry. To overcome this limitation and to provide a full range of asymptotic normal behaviour, we introduce a new \u2018complex Bingham quartic distribution\u2019 by adding a selection of quartic terms to the log-density. In the simplest case this new distribution corresponds to Kent's FB5-distribution. Asymptotic and saddlepoint methods are developed for the normalizing constant to facilitate maximum likelihood estimation. Examples are given to show the usefulness of this new distribution.<\/jats:p>","DOI":"10.1111\/j.1467-9868.2006.00565.x","type":"journal-article","created":{"date-parts":[[2006,10,24]],"date-time":"2006-10-24T15:34:04Z","timestamp":1161704044000},"page":"747-765","source":"Crossref","is-referenced-by-count":13,"title":["The Complex Bingham Quartic Distribution and Shape Analysis"],"prefix":"10.1093","volume":"68","author":[{"given":"J. T.","family":"Kent","sequence":"first","affiliation":[{"name":"University of Leeds , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"K. V.","family":"Mardia","sequence":"additional","affiliation":[{"name":"University of Leeds , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P.","family":"McDonnell","sequence":"additional","affiliation":[{"name":"Department for Environment , Food and Rural Affairs, York , UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2006,10,24]]},"reference":[{"key":"2023040708370733800_","volume-title":"Statistical Shape Analysis","author":"Dryden","year":"1998"},{"key":"2023040708370733800_","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511623059","volume-title":"Statistical Analysis of Spherical Data","author":"Fisher","year":"1987"},{"key":"2023040708370733800_","volume-title":"A probabilistic model captures local structural bias in proteins","author":"Hamelryck","year":"2005"},{"key":"2023040708370733800_","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1112\/blms\/16.2.81","volume":"16","author":"Kendall","year":"1984","journal-title":"Bull. 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