{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:19:03Z","timestamp":1760145543697,"version":"build-2065373602"},"reference-count":11,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T00:00:00Z","timestamp":1722211200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia, I.P.","award":["UIDB\/00212\/2020","UIDB\/04630\/2020","UIDB\/00297\/2020"],"award-info":[{"award-number":["UIDB\/00212\/2020","UIDB\/04630\/2020","UIDB\/00297\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The measure of sphericity for positive semi-definite matrices plays a crucial role in understanding their geometric properties, especially in high-dimensional settings. This paper introduces a robust measure of sphericity, which remains invariant under orthogonal transformations and scaling. We explore its behavior in finite-dimensional cases. Additionally, we investigate the stochastic case by considering a normal distribution, analyzing the asymptotic normality of random matrices and its implications on the convergence properties of the proposed measure.<\/jats:p>","DOI":"10.3390\/axioms13080512","type":"journal-article","created":{"date-parts":[[2024,7,29]],"date-time":"2024-07-29T12:27:43Z","timestamp":1722256063000},"page":"512","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Measuring Sphericity in Positive Semi-Definite Matrices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9095-0947","authenticated-orcid":false,"given":"D\u00e1rio","family":"Ferreira","sequence":"first","affiliation":[{"name":"Department of Mathematics and Center of Mathematics and Applications, University of Beira Interior, 6201-001 Covilha, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9209-7772","authenticated-orcid":false,"given":"Sandra S.","family":"Ferreira","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Center of Mathematics and Applications, University of Beira Interior, 6201-001 Covilha, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,29]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"254","DOI":"10.1111\/j.2517-6161.1977.tb01623.x","article-title":"Spherical Matrix Distributions and a Multivariate Model","volume":"39","author":"Dawid","year":"1977","journal-title":"J. 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Assoc."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1111\/j.1368-423X.2010.00331.x","article-title":"Testing for sphericity in a fixed effects panel data model","volume":"14","author":"Baltagi","year":"2011","journal-title":"Econom. J."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.1214\/13-AOS1100","article-title":"Asymptotic power of sphericity tests for high-dimensional data","volume":"41","author":"Onatski","year":"2013","journal-title":"Ann. Stat."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1214\/13-AOS1181","article-title":"Signal detection in high dimension: The multispiked case","volume":"42","author":"Onatski","year":"2014","journal-title":"Ann. Stat."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1093\/biomet\/ast040","article-title":"Multivariate sign-based high-dimensional tests for sphericity","volume":"101","author":"Zou","year":"2014","journal-title":"Biometrika"},{"key":"ref_9","first-page":"289","article-title":"The Eigenvalue Decomposition of a Symmetric Matrix","volume":"16","author":"Puntanen","year":"2000","journal-title":"Econom. Theory"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Kollo, T., and Von Rosen, D. (2005). Advanced Multivariate Statistics with Matrices, Springer.","DOI":"10.1007\/1-4020-3419-9"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dhrymes, P.J. (2013). Mathematicsfor Econometrics, Springer. [4th ed.].","DOI":"10.1007\/978-1-4614-8145-4"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/512\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:25:45Z","timestamp":1760109945000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/512"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,29]]},"references-count":11,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080512"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080512","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,7,29]]}}}