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Linear sufficiency and completeness in the partitioned linear model. (English) Zbl 1133.62338

Summary: We consider the estimation of \({\mathbf X}_1\beta_1\) under the partitioned linear model \(\{{\mathbf y},{\mathbf X}_1\beta_1+ {\mathbf X}_2\beta_2, \sigma^2{\mathbf V}\}\). In particular, we consider linear sufficiency and linear completeness of \({\mathbf X}_1\beta_1\). We give new characterizations for linear sufficiency of \({\mathbf X}_1\beta_1\), and define and characterize linear completeness in a case of estimation of \({\mathbf X}_1\beta_1\). We also introduce a predictive approach for obtaining the best linear unbiased estimator of \({\mathbf X}_1\beta_1\), and subsequently, we give linear analogues of the Rao-Blackwell and Lehmann-Scheffé theorems in the context of estimating \({\mathbf X}_1\beta_1\).

MSC:

62J05 Linear regression; mixed models
62B05 Sufficient statistics and fields
62H12 Estimation in multivariate analysis
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