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Transformation approaches of linear random-effects models

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  • Yongge Tian

    (Central University of Finance and Economics)

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.

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  • Yongge Tian, 2017. "Transformation approaches of linear random-effects models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(4), pages 583-608, November.
    • Handle: RePEc:spr:stmapp:v:26:y:2017:i:4:d:10.1007_s10260-017-0381-3
    DOI: 10.1007/s10260-017-0381-3
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    References listed on IDEAS

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