{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:47:56Z","timestamp":1760150876186,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,2,1]],"date-time":"2022-02-01T00:00:00Z","timestamp":1643673600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>An extension of the D2 test statistic to test the equality of mean for high-dimensional and k-th order array-variate data using k-self similar compound symmetry (k-SSCS) covariance structure is derived. The k-th order data appear in many scientific fields including agriculture, medical, environmental and engineering applications. We discuss the property of this k-SSCS covariance structure, namely, the property of Jordan algebra. We formally show that our D2 test statistic for k-th order data is an extension or the generalization of the D2 test statistic for second-order data and for third-order data, respectively. We also derive the D2 test statistic for third-order data and illustrate its application using a medical dataset from a clinical trial study of the eye disease glaucoma. The new test statistic is very efficient for high-dimensional data where the estimation of unstructured variance-covariance matrix is not feasible due to small sample size.<\/jats:p>","DOI":"10.3390\/sym14020291","type":"journal-article","created":{"date-parts":[[2022,2,1]],"date-time":"2022-02-01T22:16:18Z","timestamp":1643753778000},"page":"291","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Mean Equality Tests for High-Dimensional and Higher-Order Data with k-Self Similar Compound Symmetry Covariance Structure"],"prefix":"10.3390","volume":"14","author":[{"given":"Ricardo","family":"Leiva","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1tica, Facultad de Ciencias Econ\u00f3micas, Universidad Nacional de Cuyo, Mendoza 5500, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0294-6185","authenticated-orcid":false,"given":"Anuradha","family":"Roy","sequence":"additional","affiliation":[{"name":"Department of Management Science and Statistics, The University of Texas at San Antonio, San Antonio, TX 78249, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1007\/s11749-017-0549-z","article-title":"Testing of multivariate repeated measures data with block exchangeable covariance structure","volume":"27","author":"Klein","year":"2018","journal-title":"Test"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1080\/00949659908811970","article-title":"The mle algorithm for the matrix normal distribution","volume":"64","author":"Dutilleul","year":"1999","journal-title":"J. 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