摘要
考虑一般的Gauss—Markov模型如下M={y,Xβ,v^2V},其中y是n×1的观察值向量,它的期望是E(y)=Xβ,协方差阵是D(y)=σ~2V,这里X是n×p阶已知矩阵,β是p×1的未知参数向量,V是n×n阶已知的非负定距阵σ~2>0是未知参数.本文不须对矩阵X和V的秩作任何假设.
In this paper, we discuss the decomposition of the space μ(X : V) and the invariance with respect to the choice of a generalized inverse of matrix X in the general Gauss-Markov model. In Theorem 1, we give necessary and sufficient conditions for the least squares estimator Pxy = BLUE(Xβ) under the general Gauss-Markov model M = {y,Xβ,σ2V}. In Theorem 2, we prove that Pxy = BLUE(Xβ) under model M and invariant with respect to the choice of a generalized inverse of matrix X are equivalent.
