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Further Properties of the Linear Sufficiency in the Partitioned Linear Model

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Matrices, Statistics and Big Data (IWMS 2016)

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Abstract

A linear statistic Fy, where F is an f × n matrix, is called linearly sufficient for estimable parametric function K β under the model \({\mathcal {M}} = \{ \mathbf {y}, \mathbf {X}{\boldsymbol {\beta }}, \mathbf {V} \}\), if there exists a matrix A such that AFy is the \( \operatorname {\mathrm {BLUE}}\) for K β. In this paper we consider some particular aspects of the linear sufficiency in the partitioned linear model where X = (X 1 : X 2) with β being partitioned accordingly. We provide new results and new insightful proofs for some known facts, using the properties of relevant covariance matrices and their expressions via certain orthogonal projectors. Particular attention will be paid to the situation under which adding new regressors (in X 2) does not affect the linear sufficiency of Fy.

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Acknowledgements

Thanks go to the referees for constructive comments. Part of this research was done during the meeting of an International Research Group on Multivariate Models in the Mathematical Research and Conference Center, Bȩdlewo, Poland, March 2016, supported by the Stefan Banach International Mathematical Center.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Methods, Poznań University of Life Sciences, Poznań, Poland

    Augustyn Markiewicz

  2. Faculty of Natural Sciences, University of Tampere, Tampere, Finland

    Simo Puntanen

Authors
  1. Augustyn Markiewicz
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  2. Simo Puntanen
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Correspondence to Simo Puntanen .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Brock University, St. Catharines, ON, Canada

    S. Ejaz Ahmed

  2. Departmental Unit of Mathematics and Physics, Polytechnic Institute of Tomar, Tomar, Portugal

    Francisco Carvalho

  3. Faculty of Natural Sciences, University of Tampere, Tampere, Finland

    Simo Puntanen

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Markiewicz, A., Puntanen, S. (2019). Further Properties of the Linear Sufficiency in the Partitioned Linear Model. In: Ahmed, S., Carvalho, F., Puntanen, S. (eds) Matrices, Statistics and Big Data. IWMS 2016. Contributions to Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-17519-1_1

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