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

Augustyn Markiewicz; Simo Puntanen

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 33-50, electronic only
  • ISSN: 2391-5455

Abstract

top
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

How to cite

top

Augustyn Markiewicz, and Simo Puntanen. "All about the ⊥ with its applications in the linear statistical models." Open Mathematics 13.1 (2015): 33-50, electronic only. <http://eudml.org/doc/268783>.

@article{AugustynMarkiewicz2015,
abstract = {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},
author = {Augustyn Markiewicz, Simo Puntanen},
journal = {Open Mathematics},
keywords = {Best linear unbiased estimator; Column space; Generalized inverse; Linear statistical model; Orthocomplement; Orthogonal projector; best linear unbiased estimator; column space; generalized inverse; linear statistical model; orthocomplement; orthogonal projector},
language = {eng},
number = {1},
pages = {33-50, electronic only},
title = {All about the ⊥ with its applications in the linear statistical models},
url = {http://eudml.org/doc/268783},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Augustyn Markiewicz
AU - Simo Puntanen
TI - All about the ⊥ with its applications in the linear statistical models
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 33
EP - 50, electronic only
AB - 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
LA - eng
KW - Best linear unbiased estimator; Column space; Generalized inverse; Linear statistical model; Orthocomplement; Orthogonal projector; best linear unbiased estimator; column space; generalized inverse; linear statistical model; orthocomplement; orthogonal projector
UR - http://eudml.org/doc/268783
ER -

References

top
  1. [1] Baksalary J.K., An elementary development of the equation characterizing best linear unbiased estimators, Linear Algebra Appl., 2004, 388, 3-6 Zbl1052.62062
  2. [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 Zbl0611.62073
  3. [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] Zbl0694.15003
  4. [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 Zbl0727.62072
  5. [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] Zbl0765.62068
  6. [6] Baksalary O.M., Trenkler G., A projector oriented approach to the best linear unbiased estimator, Statist. Papers, 2009, 50, 721-733[Crossref] Zbl1247.62165
  7. [7] Baksalary O.M., Trenkler G., Between OLSE and BLUE, Aust. N. Z. J. Stat., 2011, 53, 289-303 
  8. [8] Baksalary O.M., Trenkler G., Rank formulae from the perspective of orthogonal projectors, Linear Multilinear Algebra, 2011, 59, 607-625 Zbl1220.15005
  9. [9] Baksalary O.M., Trenkler G., Liski E.P., Let us do the twist again. Statist. Papers, 2013, 54, 1109-1119[Crossref] Zbl06224849
  10. [10] Ben-Israel A., Greville T.N.E., Generalized inverses: theory and applications, 2nd Ed., Springer, New York, 2003 Zbl1026.15004
  11. [11] Ben-Israel A., The Moore of the Moore-Penrose inverse, Electron. J. Linear Algebra, 9, 150-157, 2002 Zbl1024.01012
  12. [12] Bhimasankaram P., Sengupta D., The linear zero functions approach to linear models, Sankhy¯a B, 1996, 58, 338-351 Zbl0874.62073
  13. [13] Christensen R., Plane answers to complex questions: the theory of linear models, 4th Ed. Springer, New York, 2011 Zbl1266.62043
  14. [14] Davidson R., MacKinnon J.G., Econometric theory and methods, Oxford University Press, New York, 2004 
  15. [15] Frisch R., Waugh F.V., Partial time regressions as compared with individual trends, Econometrica, 1933, 1, 387-401[Crossref] Zbl59.1207.14
  16. [16] Groß J., The general Gauss-Markov model with possibly singular dispersion matrix, Statist. Papers, 2004, 45, 311-336[Crossref] Zbl1048.62064
  17. [17] Groß J., Puntanen S., Estimation under a general partitioned linear model, Linear Algebra Appl., 2000, 321, 131-144 Zbl0966.62033
  18. [18] Groß J., Puntanen S., Extensions of the Frisch-Waugh-Lovell Theorem, Discuss. Math. Probab. Stat., 2005, 25, 39-49 Zbl1134.62302
  19. [19] Harville D.A., Matrix algebra from a statistician’s perspective, Springer, New York, 1997 Zbl0881.15001
  20. [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] Zbl1247.62167
  21. [21] Hauke J., Markiewicz A., Puntanen S., Comparing the BLUEs under two linear models, Comm. Statist. Theory Methods, 2012, 41, 2405-2418[Crossref] Zbl1319.62138
  22. [22] Herr D.G., On the history of the use of geometry in the general linear model, Amer. Statist., 1980, 34, 43-47 Zbl0434.62045
  23. [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] Zbl1102.62072
  24. [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] Zbl1163.62051
  25. [25] Kala R., Projectors and linear estimation in general linear models, Comm. Statist. Theory Methods, 1981, 10, 849-873[Crossref] Zbl0465.62060
  26. [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] Zbl0136.40704
  27. [27] Kruskal W., When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Statist., 1968, 39, 70-75[Crossref] Zbl0162.21902
  28. [28] LaMotte L.R., A direct derivation of the REML likelihood function, Statist. Papers, 2007, 48, 321-327[Crossref] Zbl1110.62078
  29. [29] Lovell M.C., Seasonal adjustment of economic time series and multiple regression analysis, J. Amer. Statist. Assoc., 1963, 58, 993-1010[Crossref] Zbl0113.14302
  30. [30] Lovell M.C., A simple proof of the FWL Theorem, J. Econ. Educ., 2008, 39, 88-91[Crossref] 
  31. [31] Margolis M.S., Perpendicular projections and elementary statistics, Amer. Statist., 1979, 33, 131-135 
  32. [32] Markiewicz A., On dependence structures preserving optimality, Statist. Probab. Lett., 2001, 53, 415-419[Crossref] Zbl0983.62041
  33. [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. [34] Marsaglia G., Styan G.P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 1974, 2, 269-292 Zbl0297.15003
  35. [35] Mitra S.K., Moore B.J., Gauss-Markov estimation with an incorrect dispersion matrix, Sankhy¯a A, 1973, 35, 139-152 Zbl0277.62044
  36. [36] Mitra S.K., Rao C.R., Projections under seminorms and generalized Moore-Penrose inverses, Linear Algebra Appl., 1974, 9, 155-167[Crossref] Zbl0296.15002
  37. [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. [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. [39] Puntanen S., Styan G.P.H., Isotalo J., Matrix tricks for linear statistical models: our personal top twenty, Springer, Heidelberg, 2011 Zbl1291.62014
  40. [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. [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 Zbl0197.15802
  42. [42] Rao C.R., Unified theory of linear estimation, Sankhy¯a A 1971, 33, 371-394. [Corrigendum (1972), 34, p. 194 and p. 477] Zbl0236.62048
  43. [43] Rao C.R., Linear statistical inference and its applications, 2nd Ed., Wiley, New York, 1973 Zbl0256.62002
  44. [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] Zbl0276.62068
  45. [45] Rao C.R., Projectors, generalized inverses and the BLUE’s, J. R. Stat. Soc. Ser. B Stat. Methodol., 1974, 336, 442-448 Zbl0291.62077
  46. [46] Rao C.R., Mitra S.K., Generalized nverse of matrices and its applications, Wiley, New York, 1971 
  47. [47] Rao C.R., Rao M.B., Matrix algebra and its applications to statistics and econometrics, World Scientific, River Edge, NJ, 1998 Zbl0915.15001
  48. [48] Searle S.R., Casella G., McCulloch C.E., Variance components, Wiley, New York, 1992 Zbl1108.62064
  49. [49] Seber G.A.F., The linear hypothesis: a general theory, 2nd Ed., Griffin, London, 1980 Zbl0418.62045
  50. [50] Seber G.A.F., Lee A.J., Linear regression analysis, 2nd Ed. Wiley, New York, 2003 Zbl1029.62059
  51. [51] Sengupta D., Jammalamadaka S.R., Linear models: an integrated approach, World Scientific, River Edge, NJ., 2003 Zbl1049.62080
  52. [52] Tian Y., On equalities for BLUEs under misspecified Gauss-Markov models, Acta Math. Sin. (Engl. Ser.), 2009, 25, 1907-1920[Crossref] Zbl1180.62083
  53. [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] Zbl1148.62053
  54. [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] Zbl1141.62043
  55. [55] Tian Y., Takane Y., On V -orthogonal projectors associated with a semi-norm, Ann. Inst. Statist. Math., 2009, 61, 517-530 [Crossref] Zbl1332.15016
  56. [56] Trenkler G., On the singularity of the sample covariance matrix, J. Stat. Comput. Simul., 1995, 52, 172-173[Crossref] 
  57. [57] Watson G.S., Serial correlation in regression analysis, I, Biometrika, 1955, 42, 327-341[Crossref] Zbl0068.33201
  58. [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] Zbl0171.17103
  59. [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 Zbl0193.47301

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.