多元线性模型中最小二乘估计的相对效率

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

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 .