This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{\beta} + \pmb{\varepsilon} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{\beta} $ and a matrix equality for the covariance matrix of the error term $ \pmb{\varepsilon} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{\beta} $ using various analytical algebraic operations of the given vectors and matrices in the model.
Citation: Yongge Tian. An effective treatment of adding-up restrictions in the inference of a general linear model[J]. AIMS Mathematics, 2023, 8(7): 15189-15200. doi: 10.3934/math.2023775
This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{\beta} + \pmb{\varepsilon} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{\beta} $ and a matrix equality for the covariance matrix of the error term $ \pmb{\varepsilon} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{\beta} $ using various analytical algebraic operations of the given vectors and matrices in the model.
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Yongge Tian. An effective treatment of adding-up restrictions in the inference of a general linear model[J]. AIMS Mathematics, 2023, 8(7): 15189-15200. doi: 10.3934/math.2023775
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