Year: 2025
Type: article
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.
Source: Sankhya A
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Tays
OpenAlex ID: W4413376861