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Let us do the twist again. (English) Zbl 1416.62386
Summary: W. Kraemer [Sankhyā, Ser. A 42, 130–131 (1980; Zbl 0485.62062)] posed the following problem: “Which are the \(y\), given \(X\) and \(V\), such that OLS and Gauss-Markov are equal?”. In other words, the problem aimed at identifying those vectors \(y\) for which the ordinary least squares (OLS) and Gauss-Markov estimates of the parameter vector \(X\beta\) coincide under the general Gauss-Markov model \(y=X\beta +u\). The problem was later called a “twist” to Kruskal’s Theorem, which provides conditions necessary and sufficient for the OLS and Gauss-Markov estimates of \(\beta\) to be equal. The present paper focuses on a similar problem to the one posed by Krämer in the aforementioned paper. However, instead of the estimation of \(\beta\), we consider the estimation of the systematic part \(X\beta\), which is a natural consequence of relaxing the assumption that \(X\) and \(V\) are of full (column) rank made by Krämer. Further results, dealing with the Euclidean distance between the best linear unbiased estimator (BLUE) and the ordinary least squares estimator (OLSE) of \(X\beta\), as well as with an equality between BLUE and OLSE are also provided. The calculations are mostly based on a joint partitioned representation of a pair of orthogonal projectors.

62J05 Linear regression; mixed models
62H12 Estimation in multivariate analysis
15A24 Matrix equations and identities
Full Text: DOI
[1] Baksalary, JK; Kala, R, A new bound for the Euclidean norm of the difference between the least squares and the best linear unbiased estimators, Ann Stat, 8, 679-681, (1980) · Zbl 0464.62055
[2] Baksalary, JK; Baksalary, OM; Szulc, T, A property of orthogonal projectors, Linear Algebra Appl, 354, 35-39, (2002) · Zbl 1025.15039
[3] Baksalary, OM; Trenkler, G, An alternative approach to characterize the commutativity of orthogonal projectors, Discuss Math Probab Stat, 28, 113-137, (2008) · Zbl 1155.15017
[4] Baksalary, OM; Trenkler, G, A projector oriented approach to the best linear unbiased estimator, Stat Pap, 50, 721-733, (2009) · Zbl 1247.62165
[5] Baksalary, OM; Trenkler, G, Functions of orthogonal projectors involving the Moore-Penrose inverse, Comput Math Appl, 59, 764-778, (2010) · Zbl 1189.15004
[6] Baksalary, OM; Trenkler, G, Between OLSE and BLUE, Aust N Z J Stat, 53, 289-303, (2011) · Zbl 1334.62106
[7] Baksalary OM, Trenkler G (2012) On projectors and some of their applications in statistics. In: Bapat RB, Kirkland S, Prasad KM, Puntanen S (eds) Lectures on matrix and graph methods. Manipal University Press, Manipal, pp 113-127
[8] Baksalary OM, Trenkler G (2013) On column and null spaces of functions of a pair of oblique projectors. Linear Multilinear Algebra. doi:10.1080/03081087.2012.731055 · Zbl 1277.15006
[9] Greville, TNE, Solutions of the matrix equation \(XAX = X\), and relations between oblique and orthogonal projectors, SIAM J Appl Math, 26, 828-832, (1974) · Zbl 0288.15018
[10] Groß, J, The general Gauss-Markov model with possibly singular dispersion matrix, Stat Pap, 45, 311-336, (2004) · Zbl 1048.62064
[11] Groß J, Trenkler G, Werner HJ (2001) The equality of linear transforms of the ordinary least squares estimator and the best linear unbiased estimator. Sankhyā 63:118-127 · Zbl 1004.62056
[12] Jaeger, A; Krämer, W, A final twist on the equality of OLS and GLS, Stat Pap, 39, 321-324, (1998) · Zbl 0899.62083
[13] Krämer W (1980) A note on the equality of ordinary least squares and Gauss-Markov estimates in the general linear model. Sankhyā 42:130-131 · Zbl 0485.62062
[14] Krämer W, Bartels R, Fiebig DG (1996) Another twist on the equality of OLS and GLS. Stat Pap 37: 277-281 · Zbl 0858.62053
[15] Kruskal, W, When are Gauss-Markov and least squares estimators identical? A coordinate-free approach, Ann Math Stat, 39, 70-75, (1968) · Zbl 0162.21902
[16] Puntanen S, Styan GPH, Isotalo J (2011) Matrix tricks for linear statistical models: our personal top twenty. Springer, Heidelberg · Zbl 1291.62014
[17] Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New York · Zbl 0162.21902
[18] Rao CR, Mitra SK (1971) Generalized inverse of matrices and its applications. Wiley, New York · Zbl 1025.15039
[19] Trenkler G (1994) Characterizations of oblique and orthogonal projectors. In: Caliński T, Kala R (eds) Proceedings of the international conference on linear statistical inference LINSTAT’93. Kluwer, Dordrecht, pp 255-270 · Zbl 0827.62054
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