{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,13]],"date-time":"2026-10-13T11:07:42Z","timestamp":1791889662877,"version":"4.3.4"},"reference-count":8,"publisher":"Oxford University Press (OUP)","issue":"3","license":[{"start":{"date-parts":[[1974,7,1]],"date-time":"1974-07-01T00:00:00Z","timestamp":141868800000},"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":[[1974,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>It is well known that in the Gauss\u2013Markov model (Y, X\u03b2, \u03c32 V) with | V | \u2260 0, the BLUE (best linear unbiased estimator) of X\u03b2 is Y1, the orthogonal projection of Y on M(X), the space spanned by the columns of X, with inner product defined as (x, y) = x'V-1 y. A quadratic function of Y2, the projection of Y on the orthogonal complement of M(X), provides an estimate of \u03c32. It may be seen that Y = Y1 + Y2. When V is singular, the inner product definition as in non-singular case is not possible. In this paper a suitable theory of projection operators is developed for the case | V | = 0, and a decomposition Y = Y1 + Y2 is obtained such that Y1 is the BLUE of X\u03b2 and a quadratic function of Y2 is the MINQUE (Minimum Norm Quadratic Unbiased Estimator) of \u03c32 in the sense of Rao (1972).<\/jats:p>","DOI":"10.1111\/j.2517-6161.1974.tb01019.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T05:10:23Z","timestamp":1544159423000},"page":"442-448","source":"Crossref","is-referenced-by-count":69,"title":["Projectors, Generalized Inverses and the Blue'S"],"prefix":"10.1093","volume":"36","author":[{"given":"C. Radhakrishna","family":"Rao","sequence":"first","affiliation":[{"name":"Indian Statistical Institute and Indiana University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090920482014200_rssb01019-cit-0001","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1017\/S0305004100032916","article-title":"Orthogonal and oblique projectors and the characteristics of pairs of vector spaces","volume":"53","author":"Afriat","year":"1957","journal-title":"Proc. Cambridge Phil. Soc."},{"key":"2024090920482014200_rssb01019-cit-0002","first-page":"57","article-title":"On the proof of the method of least squares","volume":"1","author":"Kolmogorov","year":"1946, 1947","journal-title":"Uspeki Matem, Nauk N.S."},{"key":"2024090920482014200_rssb01019-cit-0003","article-title":"Projections under seminorms and generalized inverse of matrices","volume-title":"J. Algebra","author":"Mitra","year":"1973"},{"key":"2024090920482014200_rssb01019-cit-0004","doi-asserted-by":"crossref","first-page":"152","DOI":"10.1111\/j.2517-6161.1962.tb00447.x","article-title":"A note on a generalized inverse of a matrix with applications to problems of mathematical statistics","volume":"24","author":"Rao","year":"1962","journal-title":"J. R. Statist. Soc., B"},{"key":"2024090920482014200_rssb01019-cit-0005","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1080\/01621459.1972.10481212","article-title":"Estimation of variance and covariance components in linear models","volume":"67","author":"Rao","year":"1972","journal-title":"J. Amer. Statist. Ass."},{"key":"2024090920482014200_rssb01019-cit-0006","doi-asserted-by":"crossref","DOI":"10.1002\/9780470316436","volume-title":"Linear Statistical Inference and its Applications","author":"Rao","year":"1973","edition":"2nd ed."},{"key":"2024090920482014200_rssb01019-cit-0007","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/0047-259X(73)90042-0","article-title":"Representation of best linear unbiased estimators in the Gauss\u2013Markoff model with a singular dispersion matrix","volume":"3","author":"Rao","year":"1973","journal-title":"J. Multiv. Anal."},{"key":"2024090920482014200_rssb01019-cit-0008","volume-title":"Generalized Inverse of Matrices and its Applications.","author":"Rao","year":"1971"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.2517-6161.1974.tb01019.x","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/36\/3\/442\/49096659\/jrsssb_36_3_442.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/36\/3\/442\/49096659\/jrsssb_36_3_442.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T21:16:56Z","timestamp":1725916616000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/36\/3\/442\/7027402"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1974,7,1]]},"references-count":8,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2018,12,5]]},"published-print":{"date-parts":[[1974,7,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.2517-6161.1974.tb01019.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[1974,7]]},"published":{"date-parts":[[1974,7,1]]}}}