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Permissible Covariance Structures for Simultaneous Retention of BLUEs in Small and Big Linear Models
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.
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 .