Equality of BLUEs for Full, Small, and Intermediate Linear Models Under Covariance Change, with Links to Data Confidentiality and Encryption

The necessary and sufficient condition for BLUEs of estimable functions of parameters in a linear fixed effect model being un-altered by a change in error covariance structure is due to Rao [18]. Structural insight into Rao’s condition can be gained by writing the quadratic form that is permitted to be added to the original covariance in block diagonal form. When the original full linear model is made smaller by reducing the number of regressors (which may include interactions of any order), block diagonal or diagonal matrices also provide insight into conditions for the entire set of full, small, and intermediate models each to retain their own BLUEs. The paper outlines the role that such changes in error covariance structure can play in data confidentiality and data encryption, especially when the covariance of the BLUEs is also retained. Extensions to linear mixed models and BLUPs are outlined in principle.