Transformation approaches of linear random-effects models

Yongge Tian ()
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Yongge Tian: Central University of Finance and Economics

Statistical Methods & Applications, 2017, vol. 26, issue 4, No 4, 583-608

Abstract: Abstract Assume that a linear random-effects model $$\mathbf{y}= \mathbf{X}\varvec{\beta }+ \varvec{\varepsilon }= \mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \varvec{\varepsilon }$$ y = X β + ε = X ( A α + γ ) + ε is transformed as $$\mathbf{T}\mathbf{y}= \mathbf{T}\mathbf{X}\varvec{\beta }+ \mathbf{T}\varvec{\varepsilon }= \mathbf{T}\mathbf{X}(\mathbf{A}\varvec{\alpha }+ \varvec{\gamma }) + \mathbf{T}\varvec{\varepsilon }$$ T y = T X β + T ε = T X ( A α + γ ) + T ε by pre-multiplying a given matrix $$\mathbf{T}$$ T of arbitrary rank. These two models are not necessarily equivalent unless $$\mathbf{T}$$ T is of full column rank, and we have to work with this derived model in many situations. Because predictors/estimators of the parameter spaces under the two models are not necessarily the same, it is primary work to compare predictors/estimators in the two models and to establish possible links between the inference results obtained from two models. This paper presents a general algebraic approach to the problem of comparing best linear unbiased predictors (BLUPs) of parameter spaces in an original linear random-effects model and its transformations, and provides a group of fundamental and comprehensive results on mathematical and statistical properties of the BLUPs. In particular, we construct many equalities for the BLUPs under an original linear random-effects model and its transformations, and obtain necessary and sufficient conditions for the equalities to hold.

Keywords: Linear random-effects model; Transformed model; BLUP; BLUE; Covariance matrix; Equality; 62H12; 62J05; 62J10 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10260-017-0381-3

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