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Upper Bounds for the Euclidean Distances Between the BLUEs Under the Partitioned Linear Fixed Model and the Corresponding Mixed Model
Revisiting Some Results in C.R. Rao’s Paper in Sankhyā in 1971
Abstract In 1971, in his seminal paper entitled Unified theory of linear estimation, C.R. Rao considered the properties of best linear unbiased estimators, BLUEs, in the general linear model $$ {\mathscr {M}}(\textbf{V}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}\}$$ M ( V ) = { y , X β , V } , where $$\textbf{V}$$ V refers to the covariance matrix of the observable random vector $$\textbf{y}$$ y and $$\textbf{X}$$ X is the model matrix. Both $$\textbf{X}$$ X and the nonnegative definite covariance matrix $$\textbf{V}$$ V are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of $$\textbf{V}$$ V given the class of BLUE’s of all estimable functions”. It is precisely Section 5 of Rao’s paper which is in our focus. In particular, we will take a good look at Rao’s Theorems 5.2 and 5.3 which answer the following question: Given the model $${\mathscr {M}}(\textbf{V}_{0}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}_{0} \}$$ M ( V 0 ) = { y , X β , V 0 } , how to characterize the set of all covariance matrices $$\textbf{V}$$ V such that every representation of the BLUE of $$\textbf{X}{\varvec{\beta }}$$ X β under $${\mathscr {M}}(\textbf{V}_{0})$$ M ( V 0 ) remains BLUE under $${\mathscr {M}}(\textbf{V})$$ M ( V ) . Our attempt is to provide some new insight into this problem area.
Permissible Covariance Structures for Simultaneous Retention of BLUEs in Small and Big Linear Models
Further remarks on permissible covariance structures for simultaneous retention of BLUEs in linear models
We consider the partitioned linear model M12(V0) = { y, X1β1 + X2 β2, V0 } and the corresponding small model M1(V0) = { y, X1β1 , V0 } . We define the set V1/12 of nonnegative definite matrices V such that every representation of the best linear unbiased estimator, BLUE, of μ1 = X1β1 under M12(V0) remains BLUE under M12(V) . Correspondingly, we can characterize the set V1 of matrices V such that every BLUE of μ1 = X1β1 under M1(V0) remains BLUE under M1(V). In this paper we focus on the mutual relations between the sets V1 and V1/12 .
Linear Sufficiency and Permissible Covariance Structures for Retention of BLUEs in Linear Models
How data or error covariance can change and still retain BLUEs as well as their covariance or the sum of squares of errors
SummaryMisspecification of the error covariance in linear models usually leads to incorrect inference and conclusions. We consider two linear models, and , with the same design matrix but different error covariance matrices. The conditions under which every representation of the best linear unbiased estimator (BLUE) of any estimable parametric vector under remains BLUE under have been well known since C.R. Rao's paper in 1971: Unified theory of linear estimation, Sankhyā Ser. A, Vol. 33, pp. 371–394. However, there are no previously published results on retaining the weighted sum of squares of errors (SSE) for non‐full‐rank design or error covariance matrices, and the question of when the covariance matrix of the BLUEs is also retained has been partially explored only recently. For change in any specified error covariance matrix, we provide necessary and sufficient conditions (nasc) for both BLUEs and their covariance matrix to remain unaltered and to retain this property for all submodels. We also consider nasc for SSE to be unchanged. We decompose SSE under error covariance changes, and derive nasc under which error covariance change leaves hypothesis tests for fixed‐effect deletion under normality unaltered. We also show that simultaneous retention of BLUEs and both their covariance and SSE is not possible. We outline the effects of weak and strong error covariance singularity. We provide applications (via data cloning) to maintaining data confidentiality in Official Statistics without using Confidentialised Unit Record Files (CURFs), to certain types of experimental design and to estimation of fixed parameters for linear models for single nucleotide polymorphisms (SNPs) in genetics.
Estimation and Sufficiency Under the Mixed Effects Extended Growth Curve Model with Compound Symmetry Covariance Structure
An extended growth curve model with fixed and random effects is considered. Under the assumption of multivariate normality, the maximum likelihood estimators of the fixed effects and the dispersion matrix are determined in a model with random nuisance parameters, both without any assumption on the covariance structure and under the assumption of compound symmetry. For this purpose, rules for differentiation of symmetric matrices are applied. Furthermore, when the experiments are designed in balanced complete blocks, particular symmetric matrices appear in the likelihood equations, allowing closed-form expressions for the estimators. It is also shown that the vector of sufficient statistics for the fixed effects extended growth curve model is also sufficient for the model with random nuisance parameters. The presented results are illustrated using a real data example.