Abstract
Assume that a linear regression model is written as a partitioned form. In such a case, it is quite convenient to determine the role of each subset of regressors, and to derive estimators of unknown partial parameters in the partitioned model. In this paper, we consider the relationships between the well-known ordinary least-squares estimators (OLSEs) and the best linear unbiased estimators (BLUEs) of the whole and partial mean parameter vectors in a general partitioned linear model with parameter restrictions. We first review some known results on the OLSEs and the BLUEs and their properties under general linear models. We then present a variety of necessary and sufficient conditions for OLSEs to be BLUEs under a general partitioned linear model with parameter restrictions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alalouf IS, Styan GPH (1979) Characterizations of estimability in the general linear model. Ann Stat 7:194–200
Alalouf IS, Styan GPH (1984) Characterizations of the conditions for the ordinary least squares estimator to be best linear unbiased. Chaubey YP, Dwivedi TD (Eds), Topics in Applied Statistics, Department of Mathematics, Concordia University, Montréal, pp 331–344
Baksalary JK, Kala R (1977) An extension of a rank criterion for the least squares estimator to be the best linear unbiased estimator. J Stat Plann Inference 1:309–312
Baksalary JK, Puntanen S (1990) 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
Baksalary JK, Puntanen S, Styan GPH (1990) 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. In: Styan GPH (ed) The Collected Papers of T.W. Anderson: 1943–1985. vol 2, pp 1579–1591. Wiley, New York
Baksalary OM, Trenkler G (2011) Between OLSE and BLUE. Aust N Z J Stat 53:289–303
Drygas H (1970) The coordinate-free approach to Gauss–Markov estimation. Springer, Berlin
Graybill FA (1961) An introduction to linear statistical models, vol I. McGraw-Hill, New York
Haslett SJ, Isotalo J, Liu Y, Puntanen S (2014) Equalities between OLSE, BLUE and BLUP in the linear model. Stat Pap 55:543–561
Isotalo J, Puntanen S (2009) A note on the equality of the OLSE and the BLUE of the parametric functions in the general Gauss-Markov model. Stat Pap 50:185–193
Kruskal W (1968) When are Gauss–Markov and least squares estimators identical? A coordinate free approach. Ann Math Stat 39:70–75
Liu Y (2009) On equality of ordinary least squares estimator, best linear unbiased estimator and best linear unbiased predictor in the general linear model. J Stat Plann Inference 139:1522–1529
Markiewicz A, Puntanen S (2015) All about the \(\perp \) with its applications in the linear statistical models. Open Math 13:33–50
Marsaglia G, Styan GPH (1974) Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2:269–292
Penrose R (1955) A generalized inverse for matrices. Proc Camb Phil Soc 51:406–413
Pringle RM, Rayner AA (1970) Expressions for generalized inverses of a bordered matrix with application to the theory of constrained linear models. SIAM Rev 12:107–115
Puntanen S, Styan GPH (1989) 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. Am Stat 43:153–164
Puntanen S, Styan GPH, Isotalo J (2011) Matrix tricks for linear statistical models: our personal top twenty. Springer, Berlin
Rao CR (1973) Representations of best linear unbiased estimators in the Gauss–Markoff model with a singular dispersion matrix. J Multivar Anal 3:276–292
Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York
Searle SR (1971) Linear models. Wiley, New York
Tian Y (2004) Using rank formulas to characterize equalities for Moore–Penrose inverses of matrix products. Appl Math Comput 147:581–600
Tian Y (2007) Some decompositions of OLSEs and BLUEs under a partitioned linear model. Int Stat Rev 75:224–248
Tian Y (2010) On equalities of estimations of parametric functions under a general linear model and its restricted models. Metrika 72:313–330
Tian Y, Jiang B (2016) Equalities for estimators of partial parameters under linear model with restrictions. J Multivar Anal 143:299–313
Tian Y, Wiens DP (2006) The equalities of ordinary least-squares estimators and best linear unbiased estimators for the restricted linear model. Stat Prob Lett 76:1165–1272
Tian Y, Zhang J (2011) Some equalities for estimations of partial coefficients under a general linear regression model. Stat Pap 52:911–920
Tian Y, Zhang X (2016) On connections among OLSEs and BLUEs of whole and partial parameters under a general linear model. Stat Prob Lett 112:105–112
Zhang X, Tian Y (2016) On decompositions of BLUEs under a partitioned linear model with restrictions. Stat Pap 57:345–364
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jiang, B., Sun, Y. On the equality of estimators under a general partitioned linear model with parameter restrictions. Stat Papers 60, 273–292 (2019). https://doi.org/10.1007/s00362-016-0837-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-016-0837-9