THE ESTIMATION OF THE DEVIATION BETWEEN THE LEAST SQUARES AND THE BEST LINEAR UNBIASED ESTIMATORS OF THE MEAN VECTOR IN A LINEAR MODEL

Consider the linear model. Y=Xβ+e where E（e）=0,cov（e）=σ²∑,∑≥0. It is well known that \hat\mu=X（X'X）-X'Yand μ^*=X（X'T-X）-X'T-Y are respectively the least squares and the best linear unbiased estimators of μ=Xβ, whereT=∑+XUX',U is a symmetric matrix satisfying Rank（T）=Rank（∑X） and T≥0. In this paper, we obtain that \left\|\hat\mu-\mu^*: \leqslant \frac\lambda_1-\lambda_i2 \sqrt\lambda_\lambda_2^\lambda_k\right\|Y-\hat\mu \|_2 where λ₄=ch₄（T）,i=1,2,…,n,λ₁≥…≥λ_n≥0.k=Rank（X）, \left\|\alpha^\prime\right\|_2=\left(\alpha^\prime \alpha^\prime\right)^\frac12 and the upper bounds to \left\|\operatornamecov(\hat\mu)-\operatornamecov\left(\mu^*\right)\right\|_n, \quad\left\|P T^2 P-\left(P P_2 r P\right)^2\right\| and \left\|_i\left(\cot ^+\left(\mu^*\right)\right)^\frac12 \operatornamecov(\hat\mu)\left(\cot ^+\left(\mu^*\right)\right)^\frac12\right\|_2, where \forall A \|_0=\left(\operatornametr\left(A^\prime A\right)^\frac\pi2\right)^\frac16 \cdot \delta \geqslant 1.