方差分量模型中回归系数的线性Minimax估计

考虑方差分量(混合线性)模型y=Xβ+U1ξ1+U2ξ2+…+Ukξk,这里Xn×p,Ui,n×ti为已知设计矩阵,βp×1是固定效应,iξ是ti×1随机效应向量,满足E(iξ)=0,cov(iξ)=σ2iIti,iξ都不相关.往往Uk=In,ξk=ek,即最后一项为随机误差,热β∈RP和i2σ>0(i=1,2,…,k)为未知参数.我们考虑β的可估函数Sβ,选取二次损失函数L(d,Sβ)=(d-Sβ)′(d-Sβ)∑ki=1ciσi2+β′X′Vk-1Xβ,然后在线性估计类中给出Sβ的惟一的mini max估计.

The Linear Minimax Estimator of Regression Coefficients in Mixed Linear Model

Consider the mixed linear modely=Xβ+U_1ξ_1+U_2ξ_2+…+U_kξ_k,E(ξ_i)=0,cov(ξ_i,ξ_j)=0,i≠j,cov(ξ_i,ξ_j)=σ~2_iV_i,i,j=1,2,…,k,where y is a random n×1 column vector,X_(n×p),U_(i,n×t_i) and V_i≥0 are known.β∈R~P,σ~2_i>0 are parameters,i=1,2,…,k.Choose a loss functionL(d,Sβ)=(d-Sβ)′(d-Sβ)∑ki=1c_iσ~2_i+β′X′V~(-1)_kXB.We obtain the unique minimax estimator of linear estimable functions Sβ in the class of linear estimators.