The inefficiency of least squares

Abstract

Two criteria are set up to judge the relative performance of the least squares estimator and the best linear unbiased estimator of β in the linear model y=Xβ+u, where E(u) = 0, E(uu′) = Г. The matrices X and Г are found so that the relative performance of least squares is worst. Both criteria give the same least favourable situation:when ℳ(X) is any one of the 2k manifolds ℳ(γ1±γn…γk±γn−k+1), where Гγi = fiγi and f1IJ…IJfn are fixed, ℳ(˙) denoting the subspace spanned by the columns of the relevant matrix. The case where allfi may be chosen in a preassigned interval is also discussed. The practical implications of the various results are mentioned.