{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,15]],"date-time":"2026-03-15T22:01:09Z","timestamp":1773612069397,"version":"3.50.1"},"reference-count":16,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[2019,5,13]],"date-time":"2019-05-13T00:00:00Z","timestamp":1557705600000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering & Physical Sciences Research Council","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,9,1]]},"abstract":"<jats:title>Summary<\/jats:title><jats:p>We develop a theory of covariance and concentration matrix estimation on any given or estimated sparsity scale when the matrix dimension is larger than the sample size. Nonstandard sparsity scales are justified when such matrices are nuisance parameters, distinct from interest parameters, which should always have a direct subject-matter interpretation. The matrix logarithmic and inverse scales are studied as special cases, with the corollary that a constrained optimization-based approach is unnecessary for estimating a sparse concentration matrix. It is shown through simulations that for large unstructured covariance matrices, there can be appreciable advantages to estimating a sparse approximation to the log-transformed covariance matrix and converting the conclusions back to the scale of interest.<\/jats:p>","DOI":"10.1093\/biomet\/asz014","type":"journal-article","created":{"date-parts":[[2019,3,1]],"date-time":"2019-03-01T20:14:17Z","timestamp":1551471257000},"page":"605-617","source":"Crossref","is-referenced-by-count":5,"title":["On sparsity scales and covariance matrix transformations"],"prefix":"10.1093","volume":"106","author":[{"given":"H S","family":"Battey","sequence":"first","affiliation":[{"name":"Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2019,5,13]]},"reference":[{"key":"2019091012190578300_B1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1093\/biomet\/asy011","article-title":"Robust estimation of high-dimensional covariance and precision matrices","volume":"105","author":"Avella-Medina,","year":"2018","journal-title":"Biometrika"},{"key":"2019091012190578300_B2","doi-asserted-by":"crossref","first-page":"3166","DOI":"10.3150\/16-BEJ840","article-title":"Eigen structure of a new class of covariance and inverse covariance matrices","volume":"23","author":"Battey,","year":"2017","journal-title":"Bernoulli"},{"key":"2019091012190578300_B3","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-0653-8","volume-title":"Matrix Analysis.","author":"Bhatia,","year":"1997"},{"key":"2019091012190578300_B4","doi-asserted-by":"crossref","first-page":"2577","DOI":"10.1214\/08-AOS600","article-title":"Covariance regularization by thresholding","volume":"36","author":"Bickel,","year":"2008","journal-title":"Ann. 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