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 $$ {\mathscr {M}}(\textbf{V}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>V</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , where $$\textbf{V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> refers to the covariance matrix of the observable random vector $$\textbf{y}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>y</mml:mi> </mml:math> and $$\textbf{X}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> is the model matrix. Both $$\textbf{X}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> and the nonnegative definite covariance matrix $$\textbf{V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of $$\textbf{V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> 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 $${\mathscr {M}}(\textbf{V}_{0}) = \{ \textbf{y}, \textbf{X}{\varvec{\beta }}, \textbf{V}_{0} \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , how to characterize the set of all covariance matrices $$\textbf{V}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> such that every representation of the BLUE of $$\textbf{X}{\varvec{\beta }}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> </mml:mrow> </mml:math> under $${\mathscr {M}}(\textbf{V}_{0})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>V</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> remains BLUE under $${\mathscr {M}}(\textbf{V})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>V</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Our attempt is to provide some new insight into this problem area.