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Equivalence of predictors under real and over-parameterized linear models

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  • Shengjun Gan
  • Yuqin Sun
  • Yongge Tian

Abstract

Assume that a real linear model y=Xβ+ε${\bf y}= {\bf X}\pmb {\beta }+ \pmb {\varepsilon }$ is over-parameterized as y=Xβ+Zγ+ε${\bf y}= {\bf X}\pmb {\beta }+ {\bf Z}\pmb {\gamma }+ \pmb {\varepsilon }$ by adding some new regressors Zγ${\bf Z}\pmb {\gamma }$. In such a case, results of statistical inferences of the unknown parameters β$\pmb {\beta }$ and ε$\pmb {\varepsilon }$ under the two models are not necessarily the same. This paper aims at characterizing relationships between the best linear unbiased predictors (BLUPs) of the joint vector ϕ=Kβ+Jε$\pmb {\phi }= {\bf K}\pmb {\beta }+ {\bf J}\pmb {\varepsilon }$ of the unknown parameters in the two models. In particular, we derive necessary and sufficient conditions for the BLUPs of ϕ$\pmb {\phi }$ to be equivalent under the real model and its over-parameterized counterpart.

Suggested Citation

  • Shengjun Gan & Yuqin Sun & Yongge Tian, 2017. "Equivalence of predictors under real and over-parameterized linear models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(11), pages 5368-5383, June.
  • Handle: RePEc:taf:lstaxx:v:46:y:2017:i:11:p:5368-5383
    DOI: 10.1080/03610926.2015.1100742
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    Cited by:

    1. 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.
    2. Bo Jiang & Yongge Tian, 2022. "Equivalence Analysis of Statistical Inference Results under True and Misspecified Multivariate Linear Models," Mathematics, MDPI, vol. 11(1), pages 1-16, December.
    3. Tian, Yongge & Wang, Cheng, 2017. "On simultaneous prediction in a multivariate general linear model with future observations," Statistics & Probability Letters, Elsevier, vol. 128(C), pages 52-59.

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