Pełnotekstowe zasoby PLDML oraz innych baz dziedzinowych są już dostępne w nowej Bibliotece Nauki.
Zapraszamy na


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników


2015 | 13 | 1 |

Tytuł artykułu

All about the ⊥ with its applications in the linear statistical models

Treść / Zawartość

Warianty tytułu

Języki publikacji



For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references








Opis fizyczny




  • Department of Mathematical and Statistical Methods, Pozna´n University of Life Sciences, Wojska Polskiego 28, PL-60637 Poznan, Poland
  • School of Information Sciences, FI-33014 University of Tampere, Finland


  • [1] Baksalary J.K., An elementary development of the equation characterizing best linear unbiased estimators, Linear Algebra Appl., 2004, 388, 3-6
  • [2] Baksalary J.K., Mathew T., Linear sufficiency and completeness in an incorrectly specified general Gauss-Markov model, Sankhy¯a A, 1986, 48, 169-180
  • [3] Baksalary J.K., Mathew T., Rank invariance criterion and its application to the unified theory of least squares, Linear Algebra Appl., 1990, 127, 393-401[Crossref]
  • [4] Baksalary J.K., Puntanen S., Styan G.P.H., A property of the dispersion matrix of the best linear unbiased estimator in the general Gauss-Markov model, Sankhy¯a A, 1990, 52, 279-296
  • [5] Baksalary J.K., Rao C.R., Markiewicz A., A study of the influence of the “natural restrictions” on estimation problems in the singular Gauss-Markov model, J. Statist. Plann. Inference, 1992, 31, 335-351[Crossref]
  • [6] Baksalary O.M., Trenkler G., A projector oriented approach to the best linear unbiased estimator, Statist. Papers, 2009, 50, 721-733[Crossref]
  • [7] Baksalary O.M., Trenkler G., Between OLSE and BLUE, Aust. N. Z. J. Stat., 2011, 53, 289-303
  • [8] Baksalary O.M., Trenkler G., Rank formulae from the perspective of orthogonal projectors, Linear Multilinear Algebra, 2011, 59, 607-625
  • [9] Baksalary O.M., Trenkler G., Liski E.P., Let us do the twist again. Statist. Papers, 2013, 54, 1109-1119[Crossref]
  • [10] Ben-Israel A., Greville T.N.E., Generalized inverses: theory and applications, 2nd Ed., Springer, New York, 2003
  • [11] Ben-Israel A., The Moore of the Moore-Penrose inverse, Electron. J. Linear Algebra, 9, 150-157, 2002
  • [12] Bhimasankaram P., Sengupta D., The linear zero functions approach to linear models, Sankhy¯a B, 1996, 58, 338-351
  • [13] Christensen R., Plane answers to complex questions: the theory of linear models, 4th Ed. Springer, New York, 2011
  • [14] Davidson R., MacKinnon J.G., Econometric theory and methods, Oxford University Press, New York, 2004
  • [15] Frisch R., Waugh F.V., Partial time regressions as compared with individual trends, Econometrica, 1933, 1, 387-401[Crossref]
  • [16] Groß J., The general Gauss-Markov model with possibly singular dispersion matrix, Statist. Papers, 2004, 45, 311-336[Crossref]
  • [17] Groß J., Puntanen S., Estimation under a general partitioned linear model, Linear Algebra Appl., 2000, 321, 131-144
  • [18] Groß J., Puntanen S., Extensions of the Frisch-Waugh-Lovell Theorem, Discuss. Math. Probab. Stat., 2005, 25, 39-49
  • [19] Harville D.A., Matrix algebra from a statistician’s perspective, Springer, New York, 1997
  • [20] Haslett S.J., Puntanen S., Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Statist. Papers, 2010, 51, 465-475[Crossref]
  • [21] Hauke J., Markiewicz A., Puntanen S., Comparing the BLUEs under two linear models, Comm. Statist. Theory Methods, 2012, 41, 2405-2418[Crossref]
  • [22] Herr D.G., On the history of the use of geometry in the general linear model, Amer. Statist., 1980, 34, 43-47
  • [23] Isotalo J., Puntanen S., Linear prediction sufficiency for new observations in the general Gauss-Markov model, Comm. Statist. Theory Methods, 2006, 35, 1011-1023[Crossref]
  • [24] Isotalo J., Puntanen S., Styan G.P.H., A useful matrix decomposition and its statistical applications in linear regression, Comm. Statist. Theory Methods, 2008, 37, 1436-1457[Crossref]
  • [25] Kala R., Projectors and linear estimation in general linear models, Comm. Statist. Theory Methods, 1981, 10, 849-873[Crossref]
  • [26] Khatri C.G., A note on a MANOVA model applied to problems in growth curves, Ann. Inst. Statist. Math., 1966, 18, 75-86[Crossref]
  • [27] Kruskal W., When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Statist., 1968, 39, 70-75[Crossref]
  • [28] LaMotte L.R., A direct derivation of the REML likelihood function, Statist. Papers, 2007, 48, 321-327[Crossref]
  • [29] Lovell M.C., Seasonal adjustment of economic time series and multiple regression analysis, J. Amer. Statist. Assoc., 1963, 58, 993-1010[Crossref]
  • [30] Lovell M.C., A simple proof of the FWL Theorem, J. Econ. Educ., 2008, 39, 88-91[Crossref]
  • [31] Margolis M.S., Perpendicular projections and elementary statistics, Amer. Statist., 1979, 33, 131-135
  • [32] Markiewicz A., On dependence structures preserving optimality, Statist. Probab. Lett., 2001, 53, 415-419[Crossref]
  • [33] Markiewicz A., Puntanen S., Styan G.P.H., A note on the interpretation of the equality of OLSE and BLUE, Pakistan J. Statist., 2010, 26, 127-134
  • [34] Marsaglia G., Styan G.P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 1974, 2, 269-292
  • [35] Mitra S.K., Moore B.J., Gauss-Markov estimation with an incorrect dispersion matrix, Sankhy¯a A, 1973, 35, 139-152
  • [36] Mitra S.K., Rao C.R., Projections under seminorms and generalized Moore-Penrose inverses, Linear Algebra Appl., 1974, 9, 155-167[Crossref]
  • [37] Puntanen S., Styan G.P.H., The equality of the ordinary least squares estimator and the best linear unbiased estimator (with discussion), Amer. Statist., 1989, 43, 151-161 [Commented by O. Kempthorne on pp. 161-162 and by S.R. Searle on pp. 162-163, Reply by the authors on p. 164]
  • [38] Puntanen S., Styan G.P.H., Reply [to R. Christensen (1990), R.W. Farebrother (1990), and D.A. Harville (1990)] (Letter to the Editor), Amer. Statist., 1990, 44, 192-193
  • [39] Puntanen S., Styan G.P.H., Isotalo J., Matrix tricks for linear statistical models: our personal top twenty, Springer, Heidelberg, 2011
  • [40] Rao C.R., Least squares theory using an estimated dispersion matrix and its application to measurement of signals. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability: Berkeley, California, 1965/1966, vol. 1, L.M. Le Cam and J. Neyman, eds., University of California Press, Berkeley, 355-372, 1967
  • [41] Rao C.R., A note on a previous lemma in the theory of least squares and some further results, Sankhy¯a A, 1968, 30, 259-266
  • [42] Rao C.R., Unified theory of linear estimation, Sankhy¯a A 1971, 33, 371-394. [Corrigendum (1972), 34, p. 194 and p. 477]
  • [43] Rao C.R., Linear statistical inference and its applications, 2nd Ed., Wiley, New York, 1973
  • [44] Rao C.R., Representations of best linear unbiased estimators in the Gauss-Markoff model with a singular dispersion matrix, J. Multivariate Anal., 1973, 3, 276-292[Crossref]
  • [45] Rao C.R., Projectors, generalized inverses and the BLUE’s, J. R. Stat. Soc. Ser. B Stat. Methodol., 1974, 336, 442-448
  • [46] Rao C.R., Mitra S.K., Generalized nverse of matrices and its applications, Wiley, New York, 1971
  • [47] Rao C.R., Rao M.B., Matrix algebra and its applications to statistics and econometrics, World Scientific, River Edge, NJ, 1998
  • [48] Searle S.R., Casella G., McCulloch C.E., Variance components, Wiley, New York, 1992
  • [49] Seber G.A.F., The linear hypothesis: a general theory, 2nd Ed., Griffin, London, 1980
  • [50] Seber G.A.F., Lee A.J., Linear regression analysis, 2nd Ed. Wiley, New York, 2003
  • [51] Sengupta D., Jammalamadaka S.R., Linear models: an integrated approach, World Scientific, River Edge, NJ., 2003
  • [52] Tian Y., On equalities for BLUEs under misspecified Gauss-Markov models, Acta Math. Sin. (Engl. Ser.), 2009, 25, 1907-1920[Crossref]
  • [53] Tian Y., Beisiegel, M., Dagenais E., Haines C., On the natural restrictions in the singular Gauss-Markov model, Statist. Papers, 2008, 49, 553-564[Crossref]
  • [54] Tian Y., Takane Y., Some properties of projectors associated with the WLSE under a general linear model. J. Multivariate Anal., 2008, 99, 1070-1082[Crossref]
  • [55] Tian Y., Takane Y., On V -orthogonal projectors associated with a semi-norm, Ann. Inst. Statist. Math., 2009, 61, 517-530 [Crossref]
  • [56] Trenkler G., On the singularity of the sample covariance matrix, J. Stat. Comput. Simul., 1995, 52, 172-173[Crossref]
  • [57] Watson G.S., Serial correlation in regression analysis, I, Biometrika, 1955, 42, 327-341[Crossref]
  • [58] Zyskind G., On canonical forms, non-negative covariance matrices and best and simple least squares linear estimators in linear models, Ann. Math. Statist., 1967, 38, 1092-1109[Crossref]
  • [59] Zyskind G., Martin F.B., On best linear estimation and general Gauss-Markov theorem in linear models with arbitrary nonnegative covariance structure, SIAM J. Appl. Math., 1969, 17, 1190-1202

Typ dokumentu



Identyfikator YADDA

JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.