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On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model. (English) Zbl 1341.62133
Summary: This paper presents a new investigation to the connections among the ordinary least squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) of the whole and partial mean parameter vectors in a multiple partitioned linear model. We first give some general results on the equivalence of the OLSEs and the BLUEs under a general linear model, and derive some new facts on the connections among the OLSEs and the BLUEs of the whole and partial mean parameter vectors in the model.

62H12 Estimation in multivariate analysis
62J05 Linear regression; mixed models
Full Text: DOI
[1] Aitken, A. C., On least squares and linear combination of observations, Proc. Roy. Soc. Edinburgh Sect. A, 55, 42-47, (1934) · Zbl 0011.26603
[2] Alalouf, I. S.; Styan, G. P.H., Characterizations of estimability in the general linear model, Ann. Statist., 7, 194-200, (1979) · Zbl 0398.62053
[3] Alalouf, I. S.; Styan, G. P.H., Characterizations of the conditions for the ordinary least squares estimator to be best linear unbiased, (Chaubey, Y. P.; Dwivedi, T. D., Topics in Applied Statistics, (1984), Dept. of Mathematics, Concordia Univ Montréal), 331-344
[4] Baksalary, J. K.; Kala, R., An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator, J. Statist. Plann. Inference, 1, 309-312, (1977) · Zbl 0383.62041
[5] Baksalary, J. K.; Puntanen, S., Characterizations of the best linear unbiased estimator in the general Gauss-Markov model with the use of matrix partial orderings, Linear Algebra Appl., 127, 363-370, (1990) · Zbl 0695.62152
[6] Baksalary, J. K.; Puntanen, S.; Styan, G. P.H., On T.W. anderson’s contributions to solving the problem of when the ordinary least-squares estimator is best linear unbiased and to characterizing rank additivity of matrices, (Styan, G. P.H., The Collected Papers of T.W. Anderson: 1943-1985, Vol. 2, (1990), Wiley), 1579-1591
[7] Drygas, H., The coordinate-free approach to Gauss-Markov estimation, (1970), Springer Heidelberg · Zbl 0215.26504
[8] Graybill, F. A., An introduction to linear statistical models, vol. I, (1961), McGraw-Hill · Zbl 0121.35605
[9] Guttman, L., General theory and methods for matric factoring, Psychometrika, 9, 1-16, (1944) · Zbl 0060.31212
[10] Haberman, S. J., How much do Gauss-Markov and least squares estimators differ? A coordinate free approach, Ann. Statist., 3, 982-990, (1974) · Zbl 0311.62031
[11] Haslett, S. J.; Isotalo, J.; Liu, Y.; Puntanen, S., Equalities between OLSE, BLUE and BLUP in the linear model, Statist. Papers, 55, 543-561, (2014) · Zbl 1334.62110
[12] Haslett, S. J.; Puntanen, S., A note on the equality of the BLUPs for new observations under two linear models, Acta Comment. Univ. Tartu. Math., 14, 27-33, (2010) · Zbl 1229.15023
[13] Haslett, S. J.; Puntanen, S., Equality of BLUEs or BLUPs under two linear models using stochastic restrictions, Statist. Papers, 51, 465-475, (2010) · Zbl 1247.62167
[14] Haslett, S. J.; Puntanen, S., On the equality of the BLUPs under two linear mixed models, Metrika, 74, 381-395, (2011) · Zbl 1226.62066
[15] Herzberg, A. M.; Aleong, J., Further conditions on the equivalence of ordinary least squares and weighted least squares estimators with examples, (Lanke, J.; Lindgren, G., Contributions to Probability and Statistics in Honour of Gunnar Blom, (1995), University of Lund), 127-142
[16] Isotalo, J.; Puntanen, S., A note on the equality of the OLSE and the BLUE of the parametric functions in the general Gauss-Markov model, Statist. Papers, 50, 185-193, (2009) · Zbl 1309.62113
[17] Kruskal, W., When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann. Math. Statist., 39, 70-75, (1968) · Zbl 0162.21902
[18] Marsaglia, G.; Styan, G. P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292, (1974) · Zbl 0297.15003
[19] McElroy, F. W., A necessary and sufficient condition that ordinary least-squares estimators be best linear unbiased, J. Amer. Statist. Assoc., 62, 1302-1304, (1967) · Zbl 0153.48102
[20] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51, 406-413, (1955) · Zbl 0065.24603
[21] Puntanen, S.; Styan, G. P.H., The equality of the ordinary least squares estimator and the best linear unbiased estimator, with comments by O. kempthorne, S.R. searle, and a reply by the authors, Amer. Statist., 43, 153-164, (1989)
[22] Puntanen, S.; Styan, G. P.H.; Isotalo, J., Matrix tricks for linear statistical models, our personal top twenty, (2011), Springer · Zbl 1291.62014
[23] Rao, C. R., Unified theory of linear estimation, Sankhyā Ser. A, 33, 371-394, (1971) · Zbl 0236.62048
[24] Rao, C. R., Representations of best linear unbiased estimators in the Gauss-markoff model with a singular dispersion matrix, J. Multivariate Anal., 3, 276-292, (1973) · Zbl 0276.62068
[25] Rao, C. R., Linear statistical inference and its applications, 294-302, (1973), Wiley
[26] Searle, S. R., Linear models, (1971), Wiley · Zbl 0218.62071
[27] Styan, G. P.H., When does least squares give the best linear unbiased estimate?, (Kabe, D. G.; Gupta, R. P., Multivariate Statistical Inference, (1973), North-Holland Amsterdam), 241-246 · Zbl 0264.62028
[28] Tian, Y., On equalities of estimations of parametric functions under a general linear model and its restricted models, Metrika, 72, 313-330, (2010) · Zbl 1197.62020
[29] Tian, Y., On properties of BLUEs under general linear regression models, J. Statist. Plann. Inference, 143, 771-782, (2013) · Zbl 1428.62344
[30] Tian, Y., Some equalities and inequalities for covariance matrices of estimators under linear model, Statist. Papers, (2015)
[31] Tian, Y.; Jiang, B., Equalities for estimators of partial parameters under linear model with restrictions, J. Multivariate Anal., 143, 299-313, (2016) · Zbl 1328.62347
[32] Tian, Y.; Zhang, J., Some equalities for estimations of partial coefficients under a general linear regression model, Statist. Papers, 52, 911-920, (2011) · Zbl 1229.62075
[33] 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., 17, 1190-1202, (1969) · Zbl 0193.47301
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