奇异线性模型中最小二乘估计效率的一个注记

考虑奇异线性模型:Yn×1=Xn×pβp×1+εn×1,E(ε)=0,cov(ε)=∑,设β*=(X＇∑+X)+X＇∑+Y,β=(X＇X)+X＇Y。当∑≥0和rank(X)=p时,定义最小二乘估计β与最佳线性无偏估计β*相对效率为e4(β*/β)=||cov(β*)||／||cov(β)||。当∑≥0和rank(X)＜p时,对可估函数c＇β自然考虑两种估计的方差之比的下界,提出的相对效率为e5(β*/β)=var(c＇β*)／var(c＇β)。在μ(X)(?)μ(∑)条件下,给出了它们的下界。关于相对效率的讨论通常有∑＞0的假定,利用矩阵分析的方法将协方差矩阵∑＞0推广至∑≥0的情形,从而包含了Bloomfield-Watson的结果以及推广了Kantorovich不等式。

A Note of the Relative Efficiency of the Least Squares Estimator in Singular Linear Model

The strange linear model: Yn×1 = Xn×pβp×1 + εn×1, E(ε) =0,cov(ε) = ∑. Let β* = (X'∑+ X) + X'∑+ Y , β=(X' X) + X'Y. When ∑≥0 and rank(X) = p,it is defined that the relative efficiency of least square estimator β compare to the best linear unbiased estimator β* : e4(β*/β)= ||cov(β*) || / || cov(β) || . When ∑≥0 and rank( X) < p , consider naturally lower bound of the ratio of two estimators variance for an identifiable functions c'β, the relative efficiency is posed to e5(β * /β) = var(c' β* )/var(c'β) .Under the condition ofμ(X)(?)μ(∑) ,the lower bounds is obtained. There usually are the assumption of ∑>0 on the discussions of the relative effciency in the linear model, and extend ∑>0 to ∑≥0 making use of the matrix analysis, whereby it includes Bloomfield - Watson's theorem and extend the Kantorovich inequality.