Revisiting some results in C.R. Rao's paper in Sankhya 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 M(V)={y,Xβ,V}, where V refers to the covariance matrix of the observable random vector y and X is the model matrix. Both X and the nonnegative definite covariance matrix V are known. Citing Rao, “In Section 5 [of his paper] we raise the question of identification of 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 M(V0)={y,Xβ,V0}, how to characterize the set of all covariance matrices V such that every representation of the BLUE of Xβ under M(V0) remains BLUE under M(V). Our attempt is to provide some new insight into this problem area.