线性模型中均值向量的最小二乘估计和最佳线性无偏估计之差的范数界的注记

A NOTE ON A BOUND OF THE NORM OF THE DIFFERENCE BETWEEN THE LEAST SQUEARS AND THE BEST LINEAR UNBIASED ESTIMATORS OF THE MEAN VECTOR IN A LINEAR MODEL

摘要: 我们讨论一般线性模型:Y=Xβ+e,E（e）=0,Cov（e）=σ²V,V为非负定协方差矩阵。我们知道μ=Xβ的最小二乘估计和最佳线性无偏估计分别为μ^*=X（X'X）^-X'Y和\hat\mu=X（X'T^-X）^-X'X^-Y,这里T=V+XUX',U是一个对称阵使得R（T）=R（V:X）以及T≥0。本文讨论V≥0时,\hat\mu与μ^*之差的范数界,把V>0时\hat\mu和μ^*之差在Haberman条件下的范数界推广到V≥0,且在取常用的欧氏范数时,得到使Haberman条件成立的便于应用的充要条件。本文还证明了2界的推广形式,并把3界推广到V≥0的情况。

Abstract: Consider the linear model: Y=Xβ+e, where E（e）=0, Cov=σ²V, V is a nonnegative definite matrix. It is well known that μ^*=X（X'X）^-X'Y and \hat\mu=X（X'T^-X）^-X'X^-Y are respectively the least squares and the best linear unbiased estimators of μ=Xβ, where T=V+XUX', U is a symmetric matrix 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 involved is taken as the Euclidean one, a set of necessary and suffciant conditions that is easily applicable for Haborman’s condition to be true is obtained. We prove an extended form of a bound similar to that of 2, and also extend bound 3 to that V≥0.