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

Abstract

In this article, we consider the partitioned linear model $\mathscr {M}_{12}({\textbf {V}}_{0}) = \{ {\textbf {y}}, \, {\textbf {X}}_{1}\boldsymbol{\beta }_{1} + {\textbf {X}}_{2}\boldsymbol{\beta }_{2}, \, {\textbf {V}}_{0} \}$ and the corresponding small model $\mathscr {M}_{1}({\textbf {V}}_{0}) = \{ {\textbf {y}}, \, {\textbf {X}}_{1} \boldsymbol{\beta }_{1}, \, {\textbf {V}}_{0} \} .$ Following Rao [14, Sect. 5], we can characterize the set $\mathscr {V}_{12}$ of nonnegative definite matrices ${\textbf {V}}$ such that every representation of the best linear unbiased estimator, BLUE, of $\boldsymbol{\mu } = {\textbf {X}}\boldsymbol{\beta }$ under $\mathscr {M}_{12}({\textbf {V}}_{0})$ remains BLUE under $\mathscr {M}_{12}({\textbf {V}})$. Correspondingly, we can characterize the set $\mathscr {V}_{1}$ of matrices ${\textbf {V}}$ such that every BLUE of $\boldsymbol{\mu }_{1} = {\textbf {X}}_{1}\boldsymbol{\beta }_{1}$ under $\mathscr {M}_{1}({\textbf {V}}_{0})$ remains BLUE under $\mathscr {M}_{1}({\textbf {V}})$. In the first three sections of this paper, we focus on the mutual relations between the sets $\mathscr {V}_{1}$ and $\mathscr {V}_{12}$. In Section 5, we assume that under the small model $\mathscr {M}_{1}$ the ordinary least squares estimator, OLSE, of $\boldsymbol{\mu }_{1}$ equals the $\textrm{BLUE}$ of $\boldsymbol{\mu }_{1}$ and give several characterizations for the continuation of the equality of OLSE and BLUE when more X-variables are added.