# zbMATH — the first resource for mathematics

All about the $$\bot$$ with its applications in the linear statistical models. (English) Zbl 1308.62145
Summary: For an $$n \times m$$ real matrix $$\mathbf A$$ the matrix $$\mathbf A^{\bot}$$ is defined as a matrix spanning the orthocomplement of the column space of $$\mathbf A$$, when the orthogonality is defined with respect to the standard inner product $$\langle \mathbf {x,y}\rangle = \mathbf{x'y}$$. In this paper we collect together various properties of the $$\bot$$ 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.

##### MSC:
 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis 15B99 Special matrices 15A09 Theory of matrix inversion and generalized inverses
Full Text:
##### References:
 [1] Baksalary J.K., An elementary development of the equation characterizing best linear unbiased estimators, Linear Algebra Appl., 2004, 388, 3-6 · Zbl 1052.62062 [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 · Zbl 0611.62073 [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 · Zbl 0694.15003 [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 · Zbl 0727.62072 [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 · Zbl 0765.62068 [6] Baksalary O.M., Trenkler G., A projector oriented approach to the best linear unbiased estimator, Statist. Papers, 2009, 50, 721-733 · Zbl 1247.62165 [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 · Zbl 1220.15005 [9] Baksalary O.M., Trenkler G., Liski E.P., Let us do the twist again. Statist. Papers, 2013, 54, 1109-1119 · Zbl 1416.62386 [10] Ben-Israel A., Greville T.N.E., Generalized inverses: theory and applications, 2nd Ed., Springer, New York, 2003 · Zbl 1026.15004 [11] Ben-Israel A., The Moore of the Moore-Penrose inverse, Electron. J. Linear Algebra, 9, 150-157, 2002 · Zbl 1024.01012 [12] Bhimasankaram P., Sengupta D., The linear zero functions approach to linear models, Sankhy¯a B, 1996, 58, 338-351 · Zbl 0874.62073 [13] Christensen R., Plane answers to complex questions: the theory of linear models, 4th Ed. Springer, New York, 2011 · Zbl 1266.62043 [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 · JFM 59.1207.14 [16] Groß J., The general Gauss-Markov model with possibly singular dispersion matrix, Statist. Papers, 2004, 45, 311-336 · Zbl 1048.62064 [17] Groß J., Puntanen S., Estimation under a general partitioned linear model, Linear Algebra Appl., 2000, 321, 131-144 · Zbl 0966.62033 [18] Groß J., Puntanen S., Extensions of the Frisch-Waugh-Lovell Theorem, Discuss. Math. Probab. Stat., 2005, 25, 39-49 · Zbl 1134.62302 [19] Harville D.A., Matrix algebra from a statistician’s perspective, Springer, New York, 1997 · Zbl 0881.15001 [20] Haslett S.J., Puntanen S., Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Statist. Papers, 2010, 51, 465-475 · Zbl 1247.62167 [21] Hauke J., Markiewicz A., Puntanen S., Comparing the BLUEs under two linear models, Comm. Statist. Theory Methods, 2012, 41, 2405-2418 · Zbl 1319.62138 [22] Herr D.G., On the history of the use of geometry in the general linear model, Amer. Statist., 1980, 34, 43-47 · Zbl 0434.62045 [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 · Zbl 1102.62072 [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 · Zbl 1163.62051 [25] Kala R., Projectors and linear estimation in general linear models, Comm. Statist. Theory Methods, 1981, 10, 849-873 · Zbl 0465.62060 [26] Khatri C.G., A note on a MANOVA model applied to problems in growth curves, Ann. Inst. Statist. Math., 1966, 18, 75-86 · Zbl 0136.40704 [27] Kruskal W., When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Statist., 1968, 39, 70-75 · Zbl 0162.21902 [28] LaMotte L.R., A direct derivation of the REML likelihood function, Statist. Papers, 2007, 48, 321-327 · Zbl 1110.62078 [29] Lovell M.C., Seasonal adjustment of economic time series and multiple regression analysis, J. Amer. Statist. Assoc., 1963, 58, 993-1010 · Zbl 0113.14302 [30] Lovell M.C., A simple proof of the FWL Theorem, J. Econ. Educ., 2008, 39, 88-91 [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 · Zbl 0983.62041 [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 · Zbl 0297.15003 [35] Mitra S.K., Moore B.J., Gauss-Markov estimation with an incorrect dispersion matrix, Sankhy¯a A, 1973, 35, 139-152 · Zbl 0277.62044 [36] Mitra S.K., Rao C.R., Projections under seminorms and generalized Moore-Penrose inverses, Linear Algebra Appl., 1974, 9, 155-167 · Zbl 0296.15002 [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 · Zbl 1291.62014 [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 · Zbl 0197.15802 [42] Rao C.R., Unified theory of linear estimation, Sankhy¯a A 1971, 33, 371-394. [Corrigendum (1972), 34, p. 194 and p. 477] · Zbl 0236.62048 [43] Rao C.R., Linear statistical inference and its applications, 2nd Ed., Wiley, New York, 1973 · Zbl 0256.62002 [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 · Zbl 0276.62068 [45] Rao C.R., Projectors, generalized inverses and the BLUE’s, J. R. Stat. Soc. Ser. B Stat. Methodol., 1974, 336, 442-448 · Zbl 0291.62077 [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 · Zbl 0915.15001 [48] Searle S.R., Casella G., McCulloch C.E., Variance components, Wiley, New York, 1992 · Zbl 1108.62064 [49] Seber G.A.F., The linear hypothesis: a general theory, 2nd Ed., Griffin, London, 1980 · Zbl 0418.62045 [50] Seber G.A.F., Lee A.J., Linear regression analysis, 2nd Ed. Wiley, New York, 2003 · Zbl 1029.62059 [51] Sengupta D., Jammalamadaka S.R., Linear models: an integrated approach, World Scientific, River Edge, NJ., 2003 · Zbl 1049.62080 [52] Tian Y., On equalities for BLUEs under misspecified Gauss-Markov models, Acta Math. Sin. (Engl. Ser.), 2009, 25, 1907-1920 · Zbl 1180.62083 [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 · Zbl 1148.62053 [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 · Zbl 1141.62043 [55] Tian Y., Takane Y., On V -orthogonal projectors associated with a semi-norm, Ann. Inst. Statist. Math., 2009, 61, 517-530 · Zbl 1332.15016 [56] Trenkler G., On the singularity of the sample covariance matrix, J. Stat. Comput. Simul., 1995, 52, 172-173 [57] Watson G.S., Serial correlation in regression analysis, I, Biometrika, 1955, 42, 327-341 · Zbl 0068.33201 [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 · Zbl 0171.17103 [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 · Zbl 0193.47301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.