The Legend of the Equality of OLSE and BLUE: Highlighted by C. R. Rao in 1967

Algebraic Treatments of a General Linear Model With Adding-up Restrictions

“…Hence their expressions and properties are not necessarily the same, and thus it is natural to seek possible connections between these estimation results. It is, in fact, a subject area in regression analysis is to characterize relationships between different estimation results, which has deep roots with strong statistical explanation and usefulness in the theory of linear statistical models and applications; see e.g., Markiewicz et al (2021); Tian (2010) and references therein for the background and study of this subject. Based on the analytic expressions of OLSEs and BLUEs obtained, we can consider more problems in the statistical inference of general linear regression models with adding-up restrictions.…”

“…Especially, it is natural to propose the following five equalities between the OLSEs and BLUEs under the two models in (1.1) and (1. This kind of equalities for different estimators have deep roots with strong statistical explanation and usefulness in the theory of linear statistical models and applications; see e.g., Markiewicz et al (2021); Tian (2010) and references therein for the background and study of this subject area. It is believed that this work will provide remarkable insights into intrinsic natures hidden behind the adding-up restrictions, so that the algebraic treatments presented in this note will prompt other similar studies regarding different kinds of regression models with adding-up restrictions to observable random variables under various assumptions.…”

Further remarks on permissible covariance structures for simultaneous retention of BLUEs in linear models

“…Thus the inclusion B(µ | M 12 (I)) ⊆ B(µ | M 12 (V)) can be interpreted as the equality OLSE(µ) = BLUE(µ) under M 12 (V) . For the extended review of this problem area, see Markiewicz et al [5] and the references therein like Rao [10] and Zyskind [15].…”