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设θ为p×1参数向量,T_1和T_2分别为p×1和q×1统计量,ET_1=0,ET_2=0,它们的协方差矩阵为这里σ~2未知,∑>O(即∑为正定阵).众所周知,当∑_(12)≠0时,T_1不是θ的一致最小方差无偏估计.Rao提出了θ的更好估计θ~*=T_1-∑_(12)∑_(22)~(-1)T_2,称为协方差改进估计.这里所谓“更好”是指:covθ~*=∑ _(11)-∑_(12)∑_(22)~(-1)∑_(21)≤∑_(11)=covT_1,其中A≤B表示B-A≥0.于是,θ~*在均方误差意义下改进了T_1.关于这一方面的进一步结果见文献[2,3].
Let θ be the vector of p × 1 parameters, T_1 and T_2 are the statistics of p × 1 and q × 1 respectively, ET_1 = 0, ET_2 = 0, and their covariance matrix is σ ~ 2 unknown here, Σ> O Is a positive definite matrix.) It is well known that T_1 is not unbiased estimator of uniform minimum variance for θ when Σ_ (12) ≠ 0. Rao suggests a better estimate of θ θ ~ * = T_1 -Σ_ (12) Σ_ 22) ~ (-1) T_2 is called covariance improved estimate, where “better” means covθ ~ * = Σ _ (11) -Σ_ (12) Σ_ (22) Σ_ (21) ≤Σ_ (11) = covT_1, where A≤B means BA≥0. Thus, θ ~ * improves T_1 in terms of mean square error. For further results in this respect, see [2, 3].