Abstract
A linear statistic \(\mathbf {F}\mathbf {y}\) is called linearly prediction sufficient, or shortly \({{\mathrm{BLUP}}}\)-sufficient, for the new observation \(\mathbf {y}_{*}\), say, if there exists a matrix \(\mathbf {A}\) such that \(\mathbf {A}\mathbf {F}\mathbf {y}\) is the best linear unbiased predictor, \({{\mathrm{BLUP}}}\), for \(\mathbf {y}_{*}\). We review some properties of linear prediction sufficiency that have not been received much attention in the literature and provide some clarifying comments. In particular, we consider the best linear unbiased prediction of the error term related to \(\mathbf {y}_{*}\). We also explore some interesting properties of mixed linear models including the connection between a particular extended linear model and its transformed version.
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Acknowledgements
Thanks go to Professor Xu-Qing Liu and the anonymous referees for helpful comments. Part of this research was done during the meeting of an International Research Group on Multivariate and Mixed Linear Models in the Mathematical Research and Conference Center, Bȩdlewo, Poland, November 2016, supported by the Stefan Banach International Mathematical Center.
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Isotalo, J., Markiewicz, A., Puntanen, S. (2018). Some Properties of Linear Prediction Sufficiency in the Linear Model. In: Tez, M., von Rosen, D. (eds) Trends and Perspectives in Linear Statistical Inference . Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-73241-1_8
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