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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.

MSC:
 62H12 Estimation in multivariate analysis 62J05 Linear regression; mixed models
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References:
 [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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