{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,1]],"date-time":"2022-04-01T17:40:53Z","timestamp":1648834853171},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asia Pac. J. Oper. Res."],"published-print":{"date-parts":[[2015,2]]},"abstract":"<jats:p> The reflection shapes of configurations in \u211c<jats:sup>m<\/jats:sup> with k landmarks consist of all the geometric information that is invariant under compositions of similarity and reflection transformations. By considering the corresponding Schoenberg embedding, we embed the reflection shape space into the Euclidean space of all (k - 1) by (k - 1) real symmetric matrices. In this paper, we provide a computable formula of the extrinsic mean of the reflection shapes in arbitrary dimensions. Moreover, the asymptotic analysis of the extrinsic mean of the reflection shapes is studied. By using the differentiability of spectral operators, we obtain a central limit theorem of the sample extrinsic mean of the reflection shapes. As a direct application, the two-example hypothesis test of the reflection shapes is also derived. <\/jats:p>","DOI":"10.1142\/s0217595915400059","type":"journal-article","created":{"date-parts":[[2015,2,12]],"date-time":"2015-02-12T03:43:19Z","timestamp":1423712599000},"page":"1540005","source":"Crossref","is-referenced-by-count":0,"title":["A Computable Characterization of the Extrinsic Mean of Reflection Shapes and Its Asymptotic Properties"],"prefix":"10.1142","volume":"32","author":[{"given":"Chao","family":"Ding","sequence":"first","affiliation":[{"name":"School of Mathematics, University of Southampton, National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences, Southampton SO17 1BJ, United Kingdom"}]},{"given":"Hou-Duo","family":"Qi","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Southampton, Southampton SO17 1BJ, United Kingdom"}]}],"member":"219","published-online":{"date-parts":[[2015,2,11]]},"reference":[{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmva.2009.03.007"},{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"rf4","first-page":"223","volume":"70","author":"Bhattacharya A.","year":"2008","journal-title":"Sankhy\u0101. The Indian Journal of Statistics"},{"key":"rf5","first-page":"1","volume":"33","author":"Bhattacharya R. N.","year":"2003","journal-title":"Annals of Statistics"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1214\/009053605000000093"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-93093-5"},{"key":"rf10","volume-title":"Statistical Shape Analysis","author":"Dryden I. L.","year":"1998"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1093\/biomet\/asn050"},{"key":"rf12","first-page":"215","volume":"10","author":"Fr\u00e9chet M.","year":"1948","journal-title":"Annales de l'Institut Henri Poincar\u00e9"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1007\/s10463-012-0352-2"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1006\/jmva.1998.1776"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.2307\/1426091"},{"key":"rf17","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/16.2.81"},{"key":"rf19","volume-title":"Mathematical Statistics and Data Analysis","author":"Rice J. A.","year":"2007"},{"key":"rf20","doi-asserted-by":"publisher","DOI":"10.1002\/bimj.4710360409"}],"container-title":["Asia-Pacific Journal of Operational Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0217595915400059","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T15:41:17Z","timestamp":1565106077000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0217595915400059"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,2]]},"references-count":15,"journal-issue":{"issue":"01","published-online":{"date-parts":[[2015,2,11]]},"published-print":{"date-parts":[[2015,2]]}},"alternative-id":["10.1142\/S0217595915400059"],"URL":"https:\/\/doi.org\/10.1142\/s0217595915400059","relation":{},"ISSN":["0217-5959","1793-7019"],"issn-type":[{"value":"0217-5959","type":"print"},{"value":"1793-7019","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,2]]}}}