{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T03:28:54Z","timestamp":1788233334312,"version":"build-2803163510"},"reference-count":18,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[2021,8,2]],"date-time":"2021-08-02T00:00:00Z","timestamp":1627862400000},"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":[[2022,5,25]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>We investigate estimation of a normal mean matrix under the matrix quadratic loss. Improved estimation under the matrix quadratic loss implies improved estimation of any linear combination of the columns under the quadratic loss. First, an unbiased estimate of risk is derived and the Efron\u2013Morris estimator is shown to be minimax. Next, a notion of matrix superharmonicity for matrix-variate functions is introduced and shown to have properties analogous to those of the usual superharmonic functions, which may be of independent interest. Then, it is shown that the generalized Bayes estimator with respect to a matrix superharmonic prior is minimax. We also provide a class of matrix superharmonic priors that includes the previously proposed generalization of Stein\u2019s prior. Numerical results demonstrate that matrix superharmonic priors work well for low-rank matrices.<\/jats:p>","DOI":"10.1093\/biomet\/asab025","type":"journal-article","created":{"date-parts":[[2021,4,16]],"date-time":"2021-04-16T19:21:43Z","timestamp":1618600903000},"page":"503-519","source":"Crossref","is-referenced-by-count":5,"title":["Estimation under matrix quadratic loss and matrix superharmonicity"],"prefix":"10.1093","volume":"109","author":[{"given":"T","family":"Matsuda","sequence":"first","affiliation":[{"name":"Statistical Mathematics Unit, RIKEN Center for Brain Science, 2-1 Hirosawa Wako City, Saitama 351-0198, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W E","family":"Strawderman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2021,8,2]]},"reference":[{"key":"2022052609543620600_B1","doi-asserted-by":"crossref","first-page":"2347","DOI":"10.1214\/12-EJS748","article-title":"Shrinkage estimation with a matrix loss function","volume":"6","author":"Abu-Shanab,","year":"2012","journal-title":"Electron. 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