多元线性模型中最小二乘估计的相对效率  

A Relative Efficiency of the Least Sqare Estimator in Multivariate Linear Model

作　　者：侯景臣[1] 亓旭光[1] 

机构地区：[1]辽宁石油化工大学理学院,辽宁抚顺113001

出　　处：《抚顺石油学院学报》2003年第4期84-86,共3页Journal of Fushun Petroleum Institute

摘　　要：考虑一般的多元线性模型Y_(n×k)=X_(n×p)_pB_(p×k)+e_(n×k),E(e)=0,COV(e)=V∑,其中V为已知参数矩阵,∑为已知协方差矩阵。当rank(X)<p,和∑≥0时,可估函数tr(A′B)的最小二乘估计和最佳线性无偏估计的方差分别是Var(tr(A′B))=tr(A′(X′X)^+X′∑X(X′X)^+AV)和Var(tr(A′B))=tr(A′(X′∑^+X)^+AV)。提出的相对效率为Var(tr(A′B))/Var(tr(AB)),在μ(X)μ(∑)条件下,给出了它的下界,即var(C′B)/Var(C′B)≥4λ_1λ_m/(λ_1+λ_m)~2,其中λ_1,λ_m分别是∑的最大和最小非零特征根。所使用的方法将协方差矩阵∑>0推广至∑≥0,从而包含了Haberman的结果,使之所得结果更具有一般性。A general multivariate model: Yn×k = Xn×pBp×k + en× k,E(e) = 0,COV(e) = V∑ were discussed, where V is known parameter matrix and ∑ is known covarince matrix. When rank(X)< and ∑≥0 , for an identifiable functions tr(A'B), the variances of LSE and BLUE hold Var(tr( A'B)) = tr( A'(X'X) + X'∑X(X'X) + AV) and Var( tr( A'B * )) = tr( A'(X'∑+X) + AV)respectively. The relative efficiency is Var(tr(A'B * ))/Var(tr( AB)). Under the condition μ(X)μ(2) and ∑≥0 , obtain it's lower bound, i.e Var(C'B * )/Var(C'B )≥4λ1λm/(λ+ λm)2, where λ1 ,λm are maximum and minimum nonzero eigenvalues of 2 , respectively. It makes covariance matrix ∑>0 extended to ∑≥0 in used the method. It includes Haberman's theorem, so that it is a more general results .

关 键 词：多元线性模型 最小二乘估计 最佳线性无偏估计 相对效率 

分 类 号：O212.1[理学—概率论与数理统计]