【摘要】 考虑线性模型Y =Xβ +e ,这里E(e) =0 ,Cov(e,e) =σ2 V ,V是非负定矩阵。众所周知 ,μ =Xβ的最小二乘估计和最优线性无偏估计分别为 μ =X(X′X) - X′Y和 ^μ =X(X′T-X) - X′T- Y ,这里T =V +XUX′ ,U是矩阵满足R(T) =R(VX)且T≥ 0。该文讨论V≥ 0时 μ 与 ^μ的偏差。在满足一定条件下得到相似的Haberman的一个界。在欧氏范数下 ,得到使Haberman条件成立的一个便于应用的充要条件。证明了类似于 [2 ]界的推广形式 ,并把 [3]界推广到V≥ 0。更多还原

【Abstract】 Consider the linear model: Y=Xβ+e , where E(e)=0, Cov(e,e)=σ 2V, V is nonnegative definite matrix. It is well known that μ *=X(X′X) -X′Y and =X(X′T -X) -X′T -Y are respectively the least squares and the best linear unbiased estimators of μ=Xβ, where T=V+XUX′, U is a symmetric mtrix satisfying rank(T)=rank(VX) and T≥0. In this paper, a bound similar to Haberman’s is obtained when a certain condition is satisfied. If the vector norm is taken as the Euclidean one, a set of necessary and sufficient conditions that is easily applicable for Haberman’s condition to be true are obtained. We prove an extended form of a bound similar to that of , and also extend bound to that V≥0 .更多还原