A bound for the Euclidean norm of the difference between the best linear unbiased estimator and a linear unbiased estimator

Abstract

A bound is established for the Euclidean norm of the difference between the best linear unbiased estimator and any linear unbiased estimator in the general linear model. The bound involves the spectral norm of the difference between the dispersion matrices of the two estimators, and the residual sum of squares, all evaluated at the assumed model, but is independent of the provenance of the observation vector at hand. The bound, a straightforward consequence of first principles in Gauss–Markov theory, generalizes previous results on the difference between the best linear unbiased estimator and the ordinary least-squares estimator. In a numerical example from repeated precise levelling, the bound is used to analyse the sensitivity of estimates of vertical motion to the choice of estimator.