How data or error covariance can change and still retain BLUEs as well as their covariance or the sum of squares of errors

Misspecification of the error covariance in linear models usually leads to incorrect inferenceand conclusions. We consider two linear models, A and B, with the same design matrixbut different error covariance matrices. The conditions under which every representationof the best linear unbiased estimator (BLUE) of any estimable parametric vector under Aremains BLUE under B have been well known since C.R. Rao’s paper in 1971: Unifiedtheory of linear estimation, Sankhy¯a Ser. A, Vol. 33, pp. 371–394. However, there are nopreviously published results on retaining the weighted sum of squares of errors (SSE) fornon-full-rank design or error covariance matrices, and the question of when the covariancematrix of the BLUEs is also retained has been partially explored only recently. For changein any specified error covariance matrix, we provide necessary and sufficient conditions(nasc) for both BLUEs and their covariance matrix to remain unaltered and to retain thisproperty for all submodels. We also consider nasc for SSE to be unchanged. We decomposeSSE under error covariance changes, and derive nasc under which error covariance changeleaves hypothesis tests for fixed-effect deletion under normality unaltered. We also showthat simultaneous retention of BLUEs and both their covariance and SSE is not possible. Weoutline the effects of weak and strong error covariance singularity. We provide applications(via data cloning) to maintaining data confidentiality in Official Statistics without using Confidentialised Unit Record Files (CURFs), to certain types of experimental design andto estimation of fixed parameters for linear models for single nucleotide polymorphisms(SNPs) in genetics.