{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:49:58Z","timestamp":1753890598397,"version":"3.41.2"},"reference-count":18,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2025,4,16]],"date-time":"2025-04-16T00:00:00Z","timestamp":1744761600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"<jats:p>Biadditive regression models are linear models with an additive structure for their covariance matrix. We introduce commutative conditions and derive optimal estimators, namely Best Linear Unbiased Estimators (BLUE) and Best Quadratic Unbiased Estimators (BQUE). We develop a simulation study to compare the variance components estimates obtained through the proposed approach with those derived from Analysis of Variance and Markov Chain Monte Carlo methods. This research highlights that commutative orthogonal structures in these models are highly convenient to strengthen inference.<\/jats:p>","DOI":"10.3389\/fams.2025.1379210","type":"journal-article","created":{"date-parts":[[2025,4,16]],"date-time":"2025-04-16T05:28:07Z","timestamp":1744781287000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Optimal estimators in biadditive models and their families"],"prefix":"10.3389","volume":"11","author":[{"given":"Manuela","family":"Oliveira","sequence":"first","affiliation":[]},{"given":"Eug\u00e9nio","family":"Gar\u00e7\u00e3o","sequence":"additional","affiliation":[]},{"given":"Armando","family":"Alexandre","sequence":"additional","affiliation":[]},{"given":"Joana","family":"Paulino","sequence":"additional","affiliation":[]},{"given":"Jo\u00e3o","family":"Mexia","sequence":"additional","affiliation":[]}],"member":"1965","published-online":{"date-parts":[[2025,4,16]]},"reference":[{"key":"B1","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-0318-1","volume-title":"Mixed-effects Models in S and S-PLUS","author":"Pinheiro","year":"2000"},{"volume-title":"Mixed Models: Theory and Applications With R","year":"2013","author":"Demidenko","key":"B2"},{"key":"B3","article-title":"Analysis of variance for random models","volume-title":"Unbalanced data: Theory, methods, applications, and data analysis","author":"Sahai","year":"2005"},{"key":"B4","doi-asserted-by":"publisher","first-page":"203","DOI":"10.2307\/2347446","article-title":"Statistical and computational aspects of mixed model analysis","volume":"33","author":"Dempster","year":"1984","journal-title":"Appl Stat"},{"key":"B5","doi-asserted-by":"publisher","first-page":"2021","DOI":"10.1080\/03610926.2022.2117560","article-title":"Biadditive models: commutativity and optimum estimators","volume":"53","author":"Alexandre","year":"2022","journal-title":"Commun Stat"},{"key":"B6","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1007\/978-1-4615-7397-5_27","article-title":"A characterization of best linear unbiased estimators in the general linear model","volume":"2","author":"Zmyslony","year":"1978","journal-title":"J Mathem Stat Probab Theory"},{"key":"B7","doi-asserted-by":"crossref","DOI":"10.1002\/0470866993","volume-title":"Generalized Least Squares","author":"Kariya","year":"2004"},{"key":"B8","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1098\/rspa.1965.0012","article-title":"The analysis of randomized experiments with orthogonal block structure. 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