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 2 k manifolds ℳ( γ 1 ± γ n … γ k ± γ n−k+1 ), where Г γ i = f i γ i and f 1 IJ…IJ f n are fixed, ℳ(˙) denoting the subspace spanned by the columns of the relevant matrix. The case where all f i may be chosen in a preassigned interval is also discussed. The practical implications of the various results are mentioned.