THE ESTIMATION OF THE DEVIATION BETWEEN THE LEAST SQUARES AND THE BEST LINEAR UNBIASED ESTIMATORS OF THE MEAN VECTOR IN A LINEAR MODEL
Gao D · 1992
Consider the linear model. Y=Xβ+■ where E(e)=0, cov(e)=σ~2∑, ∑≥0. It is well known that ■=X(X'X)~-X'Y and μ=X(X'T-X)~-X'T-Y are respectively the least squares and the best linear unbiased estimators of μ=Xβ, where T=∑+XUX', U is a symmetric matrix satisfying Rank(T)=Rank(∑■X) and T≥0. In this paper, we obtain that ‖■-μ‖_2≤(λ_1-λ_k)/(2(λ_1λ_k)~(1/2))‖Y-■‖_2 where λ_1=ch_i(T), i=1, 2, …, n, λ_1≥…≥λ_n≥0, k=Pank(T), ‖α‖_2=(α'α)~(1/2) and the upper bounds to ‖cov(■)-cov(μ)‖_■, ‖PT~2P-(PTP)~2‖_8 and ‖(cov~+(μ))~(1/2) cov (■) (cov~+ (μ))~(1/2)‖_■, where ‖A‖_■=(tr(A'A)~(1/2))~(1/8), s≥1.