线性模型中均值向量的LSE和BLUE的偏差

考虑线性模型Y=Xβ e，这里E（e）=0，Cov(e,e)=σ^2V,V是非负定矩阵。众所周知，u=Xβ的最小二乘估计和最优线性无偏估计分别为u=X(X‘X)^-X‘Y和u=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u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。

The Deviation between the Least Squares and the Best Linear Unbiased Estimation ofthe Mean Vector in the Linear Model

Consider the linear model: Y=Xβ+e, where E(e)=0, Cov(e,e)=σ2V, 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 2], and also extend bound 3] to that V≥0.