{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T18:53:09Z","timestamp":1779389589778,"version":"3.53.1"},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2005,4,22]],"date-time":"2005-04-22T00:00:00Z","timestamp":1114128000000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2005,6]]},"abstract":"<jats:p>In this paper we consider the estimation of the expectation vector \n<jats:italic>X<\/jats:italic>\u03b2 under the general linear model \n{<jats:italic>y<\/jats:italic>,<jats:italic>X<\/jats:italic>\u03b2,\u03c3<jats:sup>2<\/jats:sup><jats:italic>V<\/jats:italic>}. We introduce a \nnew handy representation for the rank of the difference of the covariance \nmatrices of the ordinary least squares estimator OLSE(<jats:italic>X<\/jats:italic>\u03b2) (= \n<jats:italic>Hy<\/jats:italic>, say) and the best linear unbiased estimator \nBLUE(<jats:italic>X<\/jats:italic>\u03b2) (= <jats:italic>Gy<\/jats:italic>, say). From this formula some \nwell-known conditions for the equality between <jats:italic>Hy<\/jats:italic> and <jats:italic>Gy<\/jats:italic> \nfollow at once. We recall that the equality between <jats:italic>Hy<\/jats:italic> and \n<jats:italic>Gy<\/jats:italic> can be characterized by the rank-subtractivity ordering \nbetween the covariance matrices of <jats:italic>y<\/jats:italic> and <jats:italic>Hy<\/jats:italic>. This rank \ncharacterization suggests a particular presentation for the rank of the \ndifference of the covariance matrices of <jats:italic>Hy<\/jats:italic> and <jats:italic>Gy<\/jats:italic>. We \nshow, however, that this presentation is valid if and only if the model is \nconnected.<\/jats:p>","DOI":"10.1017\/s0266466605050292","type":"journal-article","created":{"date-parts":[[2008,4,21]],"date-time":"2008-04-21T11:34:36Z","timestamp":1208777676000},"page":"659-663","source":"Crossref","is-referenced-by-count":20,"title":["THREE RANK FORMULAS ASSOCIATED WITH THE COVARIANCE MATRICES OF THE \nBLUE AND THE OLSE IN THE GENERAL LINEAR MODEL"],"prefix":"10.1017","volume":"21","author":[{"given":"Simo","family":"Puntanen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"George P.H.","family":"Styan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongge","family":"Tian","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2005,4,22]]},"reference":[{"key":"S0266466605050292_ref009","doi-asserted-by":"crossref","unstructured":"Zyskind, G. & F.B. Martin (1969)On best linear estimation and a general Gauss\u2013Markov theoremin linear models with arbitrary nonnegative covariance structure.SIAM Journal on Applied Mathematics 17,1190\u20131202.","DOI":"10.1137\/0117110"},{"key":"S0266466605050292_ref001","doi-asserted-by":"crossref","unstructured":"Baksalary, J.K. & S. Puntanen (1990)Characterizations of the best linear unbiased estimator in thegeneral Gauss\u2013Markov model with the use of matrix partialorderings.Linear Algebra and Its Applications 127,363\u2013370.","DOI":"10.1016\/0024-3795(90)90349-H"},{"key":"S0266466605050292_ref005","doi-asserted-by":"crossref","unstructured":"Puntanen, S. (1996)Some matrix results related to a partitioned singular linearmodel.Communications in Statistics: Theory and Methods 25,269\u2013279.","DOI":"10.1080\/03610929608831694"},{"key":"S0266466605050292_ref003","doi-asserted-by":"crossref","unstructured":"B\u00e9rub\u00e9, J. , R.E. Hartwig , & G.P.H. Styan (1993)On canonical correlations and the degrees of non-orthogonality inthe three-way layout. In Kameo Matusita , Madan L. Puri , & Takesi Hayakawa (eds.),Statistical Sciences and Data Analysis: Proceedings of the ThirdPacific Area Statistical Conference, pp.247\u2013252.VSP International Science Publishers BV.","DOI":"10.1515\/9783112318867-024"},{"key":"S0266466605050292_ref006","doi-asserted-by":"crossref","unstructured":"Puntanen, S. & G.P.H. Styan (1989)The equality of the ordinary least squares estimator and the bestlinear unbiased estimator (with comments by O. Kempthorne and by S.R.Searle and with \u201cReply\u201d by the authors).American Statistician 43,153\u2013164.","DOI":"10.2307\/2685065"},{"key":"S0266466605050292_ref002","doi-asserted-by":"crossref","unstructured":"Baksalary, J.K. & G.P.H. Styan (1993)Around a formula for the rank of a matrix product with somestatistical applications. In R.S. Rees (ed.),Graphs, Matrices, and Designs: Festschrift in Honor of Norman J.Pullman on His Sixtieth Birthday, Lecture Notes in Pure and AppliedMathematics 139, pp.1\u201318.Marcel Dekker.","DOI":"10.1201\/9780203719916-1"},{"key":"S0266466605050292_ref007","unstructured":"Puntanen, S. , G.P.H. Styan , & H.J. Werner (2001) Determinant of a skew-symmetric matrix:Solution 1 to Problem 00.1.1 (proposed by Steve Lawford).Econometric Theory 17, 277."},{"key":"S0266466605050292_ref004","doi-asserted-by":"crossref","unstructured":"Marsaglia, G. & G.P.H. Styan (1974)Equalities and inequalities for ranks of matrices.Linear and Multilinear Algebra 2,269\u2013292.","DOI":"10.1080\/03081087408817070"},{"key":"S0266466605050292_ref008","doi-asserted-by":"crossref","unstructured":"Tian, Y. & G.P.H. Styan (2001)Some rank equalities for idempotent and involutory matrices.Linear Algebra and Its Applications 335,101\u2013117.","DOI":"10.1016\/S0024-3795(01)00297-X"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466605050292","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,20]],"date-time":"2025-06-20T19:11:03Z","timestamp":1750446663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466605050292\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,4,22]]},"references-count":9,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2005,12]]}},"alternative-id":["S0266466605050292"],"URL":"https:\/\/doi.org\/10.1017\/s0266466605050292","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,4,22]]}}}