In a linear model where data are linearly transformed or compressed, conditions for linear sufficiency provide information about whether BLUEs of \(\textbf{X}\varvec{\beta }\) or linear combinations of \(\textbf{X}\varvec{\beta }\) (i.e., \(\textbf{K}\varvec{\beta }\) in our notation) remain unchanged. When there are changes of error covariance structure to the original model, the conditions that the BLUEs are unchanged are well known. We consider the original linear model, say \(\mathscr {A}\) , and the misspecified model \(\mathscr {B}\) , which differ only in their error covariance matrices. We explore the connections between the invariance of the linear sufficiency and the invariance of the representations of the BLUEs between \(\mathscr {A}\) and \(\mathscr {B}\) .
