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

In this article, we consider the partitioned linear model M12(V0)={y,X1β1+X2β2,V0} and the corresponding small model M1(V0)={y,X1β1,V0}. Following Rao [14, Sect. 5], we can characterize the set V12 of nonnegative definite matrices V such that every representation of the best linear unbiased estimator, BLUE, of μ= Xβ under M12(V0) remains BLUE under M12(V). Correspondingly, we can characterize the set V1 of matrices V such that every BLUE of μ1= X1β1 under M1(V0) remains BLUE under M1(V). In the first three sections of this paper, we focus on the mutual relations between the sets V1 and V12. In Section 5, we assume that under the small model M1 the ordinary least squares estimator, OLSE, of μ1 equals the BLUE of μ1 and give several characterizations for the continuation of the equality of OLSE and BLUE when more X-variables are added.