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Some remarks on fundamental formulas and facts in the statistical analysis of a constrained general linear model

Abstract Parametric regression models with certain parameter restrictions occur widely in statistical data analysis and inference. In some cases, we may wish to impose linear constraints on some subsets of fixed parameters in a given parametric regression model. This paper is concerned with some fundamental inference problems on a general linear model in which the unknown parameter vector is subject to a linear matrix equation restriction . We shall introduce the technical concepts and definitions of consistency, predictability, estimability, and formulas of the best linear unbiased predictors/best linear unbiased estimators (BLUPs/BLUEs) under a constrained general linear model (CGLM). We then show how to establish BLUPs/BLUEs of all unknown parameters in the CGLM, and present various properties of the BLUPs/BLUEs using the methodology of matrix ranks and inertias.